% WORKING TITLE: The Moduli Space of Cubic Threefolds as a Ball Quotient % Daniel Allcock, James A. Carlson, and Domingo Toledo % threefolds4.tex: 1 April 2006 % threefolds7.tex: 17 June 2006 % threefolds8.tex: 22 June 2006 % threefolds9.tex: 26 June 2006 % threefolds10.tex: 30 June 2006 % threefolds11.tex: 18 July 2006 % threefolds15.tex: 4 Aug 2006 % threefolds16.tex: 9 Aug 2006 \documentclass[12pt]{amsart} \usepackage{amssymb,amscd,amsxtra} % uncomment to print cross-reference keys %\usepackage{showkeys} % uncomment to print a checkbox by every cross-reference, to % facilitate checking of cross-references. % %\usepackage{fancybox} %\let\myref\ref %\def\ref#1{\myref{#1}\fbox{\phantom{M}}} %\overfullrule=10pt \parskip=3pt plus 1pt \raggedbottom % PAPER-SPECIFIC MACROS \newcommand{\C}{\mathbb{C}} % the complex numbers \newcommand{\R}{\mathbb{R}} % the real numbers \newcommand{\Z}{\mathbb{Z}} % the integers \newcommand{\Q}{\mathbb{Q}} % the integers \newcommand{\F}{\mathbb{F}} % finite field \newcommand{\cp}{\C P} % complex projective space \newcommand{\ch}{\C H} % complex hyperbolic space \newcommand{\E}{\mathcal{E}} % the Eisenstein integers \newcommand{\V}{\mathcal{V}} % family of fourfolds \newcommand{\T}{\mathcal{T}} % family of threefolds \newcommand{\w}{\omega} % fixed cube root of 1 \newcommand{\wbar}{{\bar\w}} % fixed cube root of 1 \newcommand{\thetabar}{{\bar\theta}} % conjuate of w-wbar \newcommand{\Hprim}{H_{0}} % primitive cohomology \newcommand{\hprim}{h_{0}} % dimension of \Hprim % set of cubic forms in x0,...,x4; takes subscripts 0, s, ss \newcommand{\forms}{\mathcal{C}} \newcommand{\framed}{\mathcal{F}} \newcommand{\moduli}{\mathcal{M}} \renewcommand{\H}{\mathcal{H}} \newcommand{\per}{g} % period map \newcommand{\PGamma}{P\Gamma} % projective monodromy group \newcommand{\D}{\Delta} % discriminant \newcommand{\wt}{\mathop{\rm wt}\nolimits} \newcommand{\id}{\mathop{\rm id}\nolimits} \newcommand{\Arg}{\mathop{\rm Arg}\nolimits} \newcommand{\dimension}{\dim} % typesetting prettiness. Looks quite strange without this. \newcommand{\rhosublambda}{\rho_{\lambda^{\vphantom{-1}}}^{\vphantom{\prime}}} % NOTATION NOT DONE WITH MACROS; FOR REFERENCE % \sigma = the branched covering transformation % \Delta = the discriminant % \eta(V) = the class of the intersection of the 4fold with a 3-plane % \Lambda(V) = the Eisenstein module of a the 4fold % ABBREVS \renewcommand{\a}{\alpha} \renewcommand{\b}{\beta} \renewcommand{\c}{\gamma} \renewcommand{\d}{\delta} \newcommand{\dlowest}{\d_{\hbox{\rm\scriptsize lowest}}} \newcommand{\e}{\varepsilon} \newcommand{\A}{\mathcal{A}} \newcommand{\B}{\mathcal{B}} \newcommand{\sset}{\subseteq} \newcommand{\isomorphism}{\cong} \newcommand{\tensor}{\otimes} \newcommand{\aut}{\mathop{\rm Aut}\nolimits} \newcommand{\Aut}{\aut} \newcommand{\Hom}{\mathop{\rm Hom}\nolimits} \newcommand{\re}{\mathop{\rm Re}\nolimits} \newcommand{\GL}{{\rm GL}} \newcommand{\SL}{{\rm SL}} \newcommand{\PGL}{{\rm PGL}} \newcommand{\Sp}{{\rm Sp}} \newcommand{\U}{{\rm U}} \newcommand{\semidirect}{\rtimes} \newcommand{\Bhat}{\widehat{B}} \newcommand{\Dhat}{\widehat{\Delta}} \DeclareMathOperator{\isom}{\hbox{\rm Isom}} \DeclareMathOperator{\diag}{\hbox{\rm diag}} \newcommand{\XX}[1]{\marginpar{#1}} % variables and ideal sheaves used in nodal and chordal degenerations \newcommand{\udot}{{\dot{u}}} \newcommand{\vdot}{{\dot{v}}} \newcommand{\wdot}{{\dot{w}}} \newcommand{\zdot}{{\dot{z}}} \newcommand{\sdot}{{\dot{s}}} \newcommand{\I}{{\mathcal{I}}} \newcommand{\J}{{\mathcal{J}}} \newcommand{\K}{{\mathcal{K}}} \newcommand{\What}{\widehat{W}} \newcommand{\phantominverse}{{\vphantom{-1}}} % theorem-like environments \theoremstyle{plain} \newtheorem{theorem}{Theorem} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{proposition}[theorem]{Proposition} \numberwithin{theorem}{section} \numberwithin{equation}{section} \numberwithin{table}{section} \numberwithin{figure}{section} % remark-like environments \theoremstyle{remark} \newtheorem*{remark}{Remark} \newtheorem*{remarks}{Remarks} \newtheorem{claim}{Claim} % An arrow to the right with a label above it \newcommand{\mapright}[1]{\ \smash{ \mathop{\longrightarrow}\limits^{#1}}\ } % an \llap that you can use inside formulas without % tex changing to textstyle inside the \hbox of \llap \def\mathllap#1{\mathchoice {\llap{$\displaystyle #1$}}% {\llap{$\textstyle #1$}}% {\llap{$\scriptstyle #1$}}% {\llap{$\scriptscriptstyle #1$}}} % the set of all #1 such that #2; makes the braces and the % middle bar big enough to accomodate everything. \def\set#1#2{\left\{\,#1\mathllap{\phantom{#2}}\mathrel{\relax}\right|\left.#2\mathllap{\phantom{#1}}\,\right\}} \def\ip#1#2{\left\langle#1\mathllap{\phantom{#2}}\right|\left.\!#2\mathllap{\phantom{#1}}\right\rangle} % for centering overfull hboxes % \def\centeroverfull#1{\setbox0=\hbox{#1}\newdimen\foo\foo=\wd0\advance\foo by -\hsize\divide\foo by 2\advance\foo by\hsize\rlap{\kern\foo\llap{\box0}}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} \title[Moduli of Cubic Threefolds]{The Moduli Space of Cubic Threefolds as a Ball Quotient} % \author{Daniel Allcock} \address{Department of Mathematics\\University of Texas at Austin\\Austin, TX 78712} \email{allcock@math.utexas.edu} \urladdr{http://www.math.utexas.edu/\textasciitilde allcock} % \author{James A. Carlson} \address{Department of Mathematics\\ University of Utah\\ Salt Lake City, UT 84112;\\ \lower\smallskipamount\hbox{Clay Mathematics Institute}\\ One Bow Street\\ Cambridge, Massachusetts 02138} \email{carlson@math.utah.edu; carlson@claymath.org} \urladdr{http://www.math.utah.edu/\textasciitilde carlson} % \author{Domingo Toledo} \address{Department of Mathematics\\University of Utah\\Salt Lake City, UT 84112} \email{toledo@math.utah.edu} \urladdr{http://www.math.utah.edu/\textasciitilde toledo} % \date{August 9, 2006} \thanks{First author partly supported by NSF grants DMS~0070930, DMS-0231585 and DMS-0245120. Second and third authors partly supported by NSF grants DMS~9900543 and DMS-0200877.} % \subjclass[2000]{Primary 32G20, Secondary 14J30} % 22E40 too? \keywords{ball quotient, period map, moduli space, cubic threefold} % \begin{abstract} The moduli space of cubic threefolds in $\cp^4$, with some minor birational modifications, is the Baily-Borel compactification of the quotient of the complex 10-ball by a discrete group. We describe both the birational modifications and the discrete group explicitly. \end{abstract} \maketitle \tableofcontents \section{Introduction} \label{sec-intro} One of the most basic facts in algebraic geometry is that the moduli space of elliptic curves, which can be realized as plane cubic curves, is isomorphic to the upper half plane modulo the action of linear fractional transformations with integer coefficients. In \cite{ACT}, we showed that there is an analogous result for cubic surfaces; the analogy is clearest when we view the upper half plane as complex hyperbolic 1-space, that is, as the unit disk. The result is that the moduli space of stable cubic surfaces is isomorphic to a quotient of complex hyperbolic 4-space by the action of a specific discrete group. This is the group of matrices which preserve the Hermitian form $\diag[-1,1,1,1,1]$ and which have entries in the ring of Eisenstein integers: the ring obtained by adjoining a primitive cube root of unity to the integers. The idea of the proof is not to use the Hodge structure of the cubic surface, which has no moduli, but rather that of the cubic threefold obtained as a triple cover of $\cp^3$ branched along the cubic surface. The resulting Hodge structures have a symmetry of order three, and the moduli space of such structures is isomorphic to complex hyperbolic 4-space $\ch^4$. This is the starting point of the proof, which relies crucially on the Clemens-Griffiths Torelli Theorem for cubic threefolds \cite{Clemens-Griffiths}. The purpose of this article is to extend the analogy to cubic threefolds. The idea is to use the period map for the cubic fourfolds obtained as triple covers of $\cp^4$ branched along the threefolds, using Voisin's Torelli theorem \cite{voisin} in place of that of Clemens and Griffiths. In this case, however, a new phenomenon occurs. There is one distinguished point in the moduli space of cubic threefolds which is a point of indeterminacy for the period map. This point is the one represented by what we call a chordal cubic, meaning the secant variety of a rational normal quartic curve in $\cp^4$. The reason for the indeterminacy is that the limit Hodge structure depends on the direction of approach to the chordal cubic locus. In fact, the limit depends \emph{only} on this direction, and so the period map extends to the blowup of the moduli space. The natural period map for smooth cubic threefolds \cite{Clemens-Griffiths} embeds the moduli space in a period domain for Hodge structures of weight three, namely, a quotient of the Siegel upper half space of genus five. For this embedding, however, the target space has dimension greater than that of the source. For the construction of this article, the dimensions of the source and target are the same. To formulate the main result, let $\moduli_{ss}$ be the GIT moduli space of cubic threefolds, and let $\widehat\moduli_{ss}$ be its blowup at the point corresponding to the chordal cubics. Let $\moduli_s\sset\moduli_{ss}$ be the moduli space of stable cubic threefolds, and let $\widehat\moduli_s$ be $\widehat\moduli_{ss}$ minus the proper transform of $\moduli_{ss}-\moduli_s$. Let $\moduli_0$ be the moduli space of smooth cubic threefolds. Then we have the following, contained in the statement of the main result, theorem~\ref{thm-MAIN-THEOREM}: \begin{theorem} There is an arithmetic group $P\Gamma$ acting on complex hyperbolic 10-space, such that the period map \[ \widehat\moduli_{s} \longrightarrow P\Gamma\backslash\ch^{10} \] is a isomorphism. This map identifies $\moduli_s$ with the image in $P\Gamma\backslash\ch^{10}$ of the complement of a hyperplane arrangement $\H_c$. It also identifies the discriminant in $\widehat\moduli_{s}$ with the image of another hyperplane arrangement $\H_{\D}\sset\ch^{10}$. In particular, it identifies $\moduli_0$ with $P\Gamma\backslash\bigl(\ch^{10}-(\H_c\cup\H_{\D})\bigr)$. Finally, the period map extends to a morphism from $\widehat\moduli_{ss}$ to the Baily-Borel compactification $\overline{\PGamma\backslash\ch^{10}}$. \end{theorem} \noindent We also provide much more detailed information about all the objects in the theorem, such as explicit descriptions of $P\Gamma$, $\H_c$ and $\H_{\D}$, and an analysis of the part of $\widehat\moduli_{ss}$ lying over the boundary points of $\overline{P\Gamma\backslash\ch^{10}}$. Now we will say what the group $P\Gamma$ is. Let $V$ be a cyclic cubic fourfold, meaning a triple cover of $P^4$ branched over a cubic threefold. The primitive cohomology $\Lambda(V)$ of $V$ is naturally a module for the Eisenstein integers, where a primitive cube root of unity acts on cohomology as does the corresponding deck transformation. When $V$ is smooth, This Eisenstein module carries a natural % nondegenerate Hermitian form of signature $(10,1)$, and $P\Gamma$ is the projective isometry group of $\Lambda(V)$. The architecture of our proof of theorem~\ref{thm-MAIN-THEOREM} dates back to before \cite{allcock-threefolds}, and follows the pattern laid out in \cite{ACT} for cubic surfaces. But it is considerably more technical in its details. We therefore focus on the points where there are major differences or where substantially more work must be done as compared with the case of cubic surfaces. In section~\ref{sec-smoothmoduli} we establish basic facts about $\Lambda(V)$ as an Eisenstein module endowed with a complex Hodge structure, give an inner product matrix for $\Lambda(V)$, and show that $\moduli_0\to\PGamma\backslash\ch^{10}$ is an isomorphism onto its image. The argument here follows that of \cite{ACT}, except that in place of the Clemens-Griffiths theorem, we use Claire Voisin's Torelli theorem for cubic fourfolds \cite{voisin}. In this section we also establish facts about the discriminant locus for cubic threefolds near stable singularities. These facts are used in section~\ref{sec-extension} for extending the period map. In section~\ref{sec-discr-near-chordal-cubic} we blow up the space $P^{34}$ of cubic threefolds along the chordal cubic locus, and describe the proper transform of the discriminant locus. This is one of the most technical sections in the paper, but it is required for the extension of the period map in section~\ref{sec-extension}. To give an idea of the main issue, consider a one-parameter family of smooth cubic threefolds degenerating to a chordal cubic. We may write it as $F + tG=0$, where $F = 0$ defines the chordal cubic. The polynomial $G = 0$ cuts out on the rational normal curve of $F = 0$ a set of twelve points. Thus to a tangent vector at a point of the chordal locus one associates a 12-tuple on the projective line. We show that the discriminant locus in the blown-up $P^{34}$ has a local product structure, e.g., it is homeomorphic to the product of the discriminant locus for 12-tuples in $P^1$, times a disk representing the transverse direction, times a twenty-one dimensional ball corresponding to the action of the projective group $\PGL(5,\C)$. There is also a technical variation on this result which gives an analytic model of the discriminant, not just a topological one. In section~\ref{sec-extension} we extend the period map to the semistable locus of the blown-up $P^{34}$. This requires some geometric invariant theory to say what the semistable locus is, and here the work of Reichstein \cite{reichstein} is essential. Then we study the local monodromy groups at points in this semistable locus. The essential result for the extension of the period map is that these groups are all finite or virtually unipotent. In section~\ref{sec-hodge-theory-chordal} we show that the extended period map sends the chordal cubic locus to a divisor in $\PGamma\backslash\ch^{10}$. The main point here is to identify the limit Hodge structure and in so doing show that the derivative of the extended map along the blowup of the chordal cubic locus is of rank nine. We do this by establishing an isogeny between the limit Hodge structure and the sum of a 1-dimensional Hodge structure and a Hodge structure associated to a six-fold cover of the projective line branched at twelve points. These Hodge structures first arose in the work of Deligne and Mostow \cite{Deligne-Mostow}. Our analysis shows that the image of the period map is the quotient of a totally geodesic $\ch^9$ by a suitable subgroup of $\Gamma$. Section~\ref{sec-hodge-theory-nodal} deals with the same issues for the divisor of nodal cubic threefolds. Here the analysis is easier. We show that the Hodge structure is isomorphic to that of a special K3 surface, plus a 1-dimensional summand. This K3 surface and its Hodge structure were treated in Kond\=o's work \cite{kondo:genus-4} on moduli of genus~4 curves. In section~\ref{sec-main-theorem} we assemble the various pieces to prove the main theorem, and in section~\ref{sec-group-and-arrangement} we give some supplemental results on the monodromy group and the hyperplane arrangements. Another proof of the main theorem has been obtained by Looijenga and Swierstra \cite{looijenga-period} . Both proofs proceed by extending the period map from the moduli space of smooth threefolds to a larger space, but the extension process is quite different in the two proofs. We use a detailed analysis of the discriminant in the space obtained by blowing up the chordal cubic locus to extend the period map. Looijenga and Swierstra use a general machinery developed earlier by them \cite{looijenga-open} to handle extensions of period mappings. We are grateful to Looijenga for sending us an early version of their argument. It is a pleasure to thank the many people whose conversations have played a helpful role in the long gestation of this paper, including Herb Clemens, Alessio Corti, Johan de Jong, J\'anos Koll\'ar, Eduard Looijenga and Madhav Nori. We are grateful to the Clay Mathematics Institute for its hospitality. %We need to decide whether we want to describe the moduli space of %cubic surfaces as a suitable sub-ball quotient, and whether we want to %treat the moduli space of cubic threefolds equipped with a level 3 %structure as a quotient by a subgroup of $P\Gamma$. ($\Gamma$ acts on %$\Lambda/\theta \Lambda$, which admits an antisymmetric pairing, with %1-dimensional kernel, and the quotient is a 10-dimensional symplectic %space over $\F_3$. $\Gamma$ acts as the full symplectic group; this %is presumably the same as the monodromy on level 3 structures on the %intermediate Jacobian.) \section{Moduli of smooth cubic threefolds} \label{sec-smoothmoduli} This section contains a number of foundational results, and its main theorem is of interest in its own right. We consider cyclic cubic fourfolds, i.e., triple covers of $\cp^4$ branched along cubic threefolds. (1) The cohomology of such a fourfold is a module over $\E=\Z[\w{=}{\root3\of1}]$, equipped with a Hermitian form. (2) The monodromy on this lattice as the threefolds acquire a node is a complex reflection of order three; see Lemma \ref{lem-meridians-act-by-complex-reflections}. (3) To analyze the local monodromy for more complicated singularities, we give a structure theorem for the discriminant locus of the space of cubic threefolds near a cubic threefold with singularities of type $A_n$ and $D_4$. See Lemma \ref{lem-discriminant-away-from-E}. Using this result, we give an inner product matrix for the Hermitian form; see Lemma \ref{thm-inner-product-matrix}. (4) With the previous results in hand, we define a framing of the Hodge structure of a cyclic cubic fourfold and define the period map. Finally, the main theorem of the section is that the period map for smooth cubic threefolds is an isomorphism onto its image; see Theorem \ref{thm-main-theorem-smooth-case}. Let $\forms$ be the space of all nonzero cubic forms in variables $x_0,\dots,x_4$. For such a form $F$ let $T$ be the cubic threefold in $\cp^4$ it defines, and let $V$ be the cubic fourfold in $\cp^5$ defined by $F(x_0,\dots,x_4)+x_5^3=0$. %\XX{$x_5$ vs. $z$} Whenever we consider $F\in\forms$, $T$ and $V$ will have these meanings. $V$ is the triple cover of $\cp^4$ branched along $T$. We write $\forms_0$ for the set of $F\in\forms$ for which $T$ is smooth (as a scheme) and $\D$ for the discriminant $\forms-\forms_0$. We will sometimes also write $\D$ for its image in $P\forms$; context will make our meaning clear. We write $\forms_s$ for the set of $F\in\forms$ for which $T$ is stable in the sense of geometric invariant theory. By \cite{allcock-threefolds} or \cite{yokoyama}, this holds if and only if $T$ has no singularities of types other than $A_1,\dots,A_4$. $\forms_s$ will play a major role in sections~\ref{sec-extension}--\ref{sec-main-theorem}; in this section all we will use is the fact that $\forms_0$ lies within it. Because we will vary our threefolds, we will need the universal family $\T\sset\forms\times\cp^4$ of cubic threefolds, $$ \T=\set{(F,[x_0{:}\dots{:}x_4])\in\forms\times\cp^4}{F(x_0,\dots,x_4)=0}\;, $$ and the family of covers of $\cp^4$ branched over them, $$ \V=\set{(F,[x_0{:}\dots{:}x_5])\in\forms\times\cp^5}{F(x_0,\dots,x_4)+x_5^3=0}\;. $$ We will write $\pi_\T$ and $\pi_\V$ for the projections $\T\to\forms$ and $\V\to\forms$. The total spaces of $\T$ and $\V$ are smooth. We write $\T_0$ and $\V_0$ for the topologically locally trivial fibrations which are the restrictions of $\T$ and $\V$ to $\forms_0$. The transformation $\sigma:\C^6\to\C^6$ given by \begin{equation} \label{eq-def-of-sigma} \sigma(x_0,\dots,x_5)=(x_0,\dots,x_4,\w x_5)\;, \end{equation} where $\w$ is a fixed primitive cube root of unity, plays an essential role in all that follows. We regard it as a symmetry of $\V$ and of the individual $V$'s. Our period map $\forms_0\to\ch^{10}$ will be defined using the Hodge structure of the fourfolds and its interaction with $\sigma$, so we need to discuss $H^4(V)$ for $F\in\forms_0$. To compute this it suffices by the local triviality of $\V_0$ to consider the single fourfold $x_0^3+\dots+x_5^3=0$; by thinking of it as an iterated branched cover, one finds that its Euler characteristic is~$27$. The Lefschetz hyperplane theorem and Poincar\'e duality imply that $H^i(V;\Z)$ is the same as $H^i(\cp^5;\Z)$ for $i\neq4$, so $H^4(V;\Z)\isomorphism\Z^{23}$. The class of a 3-plane in $\cp^5$ pulls back to a class $\eta(V)\in H^4(V;\Z)$ of norm~$3$. The primitive cohomology $\Hprim^4(V;\Z)$ is the orthogonal complement of $\eta(V)$ in $H^4(V;\Z)$. Since $H^4(V;\Z)$ is a unimodular lattice, $\Hprim^4(V;\Z)$ is a $22$-dimensional lattice with determinant equal to that of $\langle\eta(V)\rangle$, up to a sign, so $\det\Hprim^4(V;\Z)=\pm3$. $\Hprim^4(V;\Z)$ is a module not only over $\Z$ but over the Eisenstein integers $\E=\Z[\w]$ as well. To see this, observe that the isomorphism $V/\langle\sigma\rangle\isomorphism \cp^4$ implies that $H^4(\cp^4;\C)$ is the $\sigma$-invariant part of $H^4(V;\C)$. Therefore $\sigma$ fixes no element of $H^4_0(V;\C)$ except $0$, hence no element of $H^4_0(V;\Z)$ except $0$. We define $\Lambda(V)$ to be the $\E$-module whose underlying additive group is $\Hprim^4(V;\Z)$, with the action of $\w\in\E$ defined as $\sigma^*$. $\E$ is a unique factorization domain, so $\Lambda(V)$ is free of rank~$11$. We define a Hermitian form on $\Lambda(V)$ by the formula \begin{equation} \label{eq-defn-of-hermitian-form} \ip{\a}{\b}=\frac{1}{2}\Bigl[ 3\a\cdot\b-\theta\a\cdot({\sigma^*}^{-1}\b-\sigma^*\b) \Bigr]\;, \end{equation} where the dot denotes the usual pairing $\a\cdot\b=\int_V\a\wedge\b$ and $\theta=\w-\wbar=\sqrt{-3}$. The scale factor $1/2$ is chosen so that $\ip{\a}{\b}$ takes values in $\E$; it is the smallest scale for which this is true. \begin{lemma} \label{lem-basic-properties-of-hermitian-pairing} $\ip{{\cdot}}{{\cdot}}$ is an $\E$-valued Hermitian form, linear in its first argument and antilinear in its second. Furthermore, $\ip{\a}{\b}\in\theta\E$ for all $\a,\b\in \Lambda(V)$. \end{lemma} \begin{proof} $\Z$-bilinearity is obvious. $\E$-linearity in its first argument holds by the following calculation. (Throughout the proof we write $\sigma$ for $\sigma^*$.) \begin{equation*} \begin{split} \ip{\theta\a}{\b} &=\ip{\sigma\a-\sigma^{-1}\a}{\b}\\ % &=\frac{1}{2}\Bigl[ 3(\sigma\a-\sigma^{-1}\a)\cdot\b-\theta(\sigma\a-\sigma^{-1}\a)\cdot(\sigma^{-1}\b-\sigma\b)\Bigr]\\ % &=\frac{\theta}{2}\Bigl[ \thetabar\sigma\a\cdot\b-\thetabar\sigma^{-1}\a\cdot\b\\ % &\mathrel{\phantom{=}}\phantom{\frac{\theta}{2}\Bigl[} -(\sigma\a\cdot\sigma^{-1}\b-\sigma\a\cdot\sigma\b -\sigma^{-1}\a\cdot\sigma^{-1}\b+\sigma^{-1}\a\cdot\sigma\b)\Bigr]\\ % &=\frac{\theta}{2}\Bigl[ \thetabar\a\cdot\sigma^{-1}\b-\thetabar\a\cdot\sigma\b\\ % &\mathrel{\phantom{=}}\phantom{\frac{\theta}{2}\Bigl[} -(\a\cdot\sigma^{-2}\b-\a\cdot\b-\a\cdot\b+\a\cdot\sigma^2\b)\Bigr]\\ % &=\frac{\theta}{2}\Bigl[ 2\a\cdot\b-\a\cdot(\sigma^{-2}\b+\sigma^2\b) -\theta(\a\cdot\sigma^{-1}\b-\a\cdot\sigma\b)\Bigr]\\ % &=\frac{\theta}{2}\Bigl[ 3\a\cdot\b-\theta\a\cdot(\sigma^{-1}\b-\sigma\b)\Bigr]\\ % &=\theta\ip{\a}{\b}\;. \end{split} \end{equation*} In the second-to-last step we used the relation $\sigma^{-2}+\sigma^2=\sigma+\sigma^{-1}=-1$. That $\ip{{\cdot}}{{\cdot}}$ is a $\C$-valued Hermitian form now follows from $\ip{\a}{\b}=\overline{\ip{\b}{\a}}$; to prove this it suffices to check that the imaginary part of \eqref{eq-defn-of-hermitian-form} changes sign when $\a$ and $\b$ are exchanged, i.e., \begin{equation*} \begin{split} \a\cdot(\sigma^{-1}\b-\sigma\b) &=\a\cdot\sigma^{-1}\b-\a\cdot\sigma\b\\ &=\sigma\a\cdot\b-\sigma^{-1}\a\cdot\b\\ &=-(\sigma^{-1}\a-\sigma\a)\cdot\b\\ &=-\b\cdot(\sigma^{-1}\a-\sigma\a)\;. \end{split} \end{equation*} Next we check that $\ip{{\cdot}}{{\cdot}}$ is $\E$-valued. It is obvious that its real part takes values in $\frac{1}{2}\Z$ and that its imaginary part takes values in $\frac{\theta}{2}\Z$. Since $$ \E=\set{a/2+b\theta/2}{\hbox{$a,b\in\Z$ and $a\equiv b$ mod 2}}\;, $$ it suffices to prove that $$ 3\a\cdot\b\equiv -\a\cdot(\sigma^{-1}\b-\sigma\b)\qquad\hbox{(mod 2)}, $$ that is, that $2$ divides $\a\cdot(\b-\sigma\b+\sigma^{-1}\b)$. This follows from the relation $1-\w+\wbar=-2\w$ in $\E$. Furthermore, \eqref{eq-defn-of-hermitian-form} shows that $\ip{\a}{\b}$ has real part in $\frac{3}{2}\Z$; since every element of $\E$ with real part in $\frac{3}{2}\Z$ is divisible by $\theta$, the proof is complete. \end{proof} We will describe $\Lambda(V)$ more precisely later, after considering the Hodge structure of $V$. By the Griffiths residue calculus, $\Hprim^{p,q}(V;\C)$ is spanned for $p+q=4$ by the residues of rational differential forms \begin{equation} \label{eq-Omega(A)} \Omega(A)=\frac{A(x_0,\dots,x_5)\Omega}{\bigl(F(x_0,\dots,x_4)+x_5^3\bigr)^{q+1}} \end{equation} where $$ \Omega=\sum_{j=0}^5(-1)^jX_j\, dX_0\wedge\dots\wedge\widehat{dX_j}\wedge\dots\wedge dX_5 $$ has degree~6 and $A$ is any homogeneous polynomial of degree $3q-3$, so that the total degree of $\Omega(A)$ is~0. Two such polynomials define the same cohomology class, modulo $\oplus_{p'>p}H^{p',4-p'}_0$, if and only if $A-A'$ lies in the Jacobian ideal of $F+x_5^3$. This gives Hodge numbers $\hprim^{4,0}=\hprim^{0,4}=0$, $\hprim^{3,1}=\hprim^{1,3}=1$ and $\hprim^{2,2}=20$. Since $\sigma$ is holomorphic, its eigenspace decomposition refines the Hodge decomposition. In particular, the generator $\Omega(1)=\Omega/(F+x_5^3)^2$ of $H^{3,1}$ has eigenvalue $\w$. Therefore $H^4_\w(V;\C)=H^{3,1}_\w\oplus H^{2,2}_\w$, the summands being one- and ten-dimensional. On $H^4(V;\C)$ there is the Hodge-theoretic Hermitian pairing \begin{equation} \label{eq-defn-of-Hodge-theoretic-pairing} (\a,\b)=3\int_V \a\wedge\bar\b\;. \end{equation} The Hodge-Riemann bilinear relations \cite[p.~123]{griffiths-harris} show that $({\cdot},{\cdot})$ is positive-definite on $H_0^{2,2}$ and negative-definite on $H^{3,1}$. It follows that $H^4_\w(V;\C)$ has signature $(10,1)$. If $W$ is a complex vector space of dimension $n+1$ with a Hermitian form of signature $(n,1)$ then we write $\ch(W)$ for the space of lines in $W$ on which the given form is negative definite. We call this the complex hyperbolic space of $W$; it is an open subset of $PW$ and is biholomorphic to the unit ball in $\C^n$. The previous two paragraphs may be summarized by saying that the Hodge structure of $V$ defines a point in $\ch\bigl(H^4_\w(V;\C)\bigr)$. We chose the factor $3$ in \eqref{eq-defn-of-Hodge-theoretic-pairing} so that $({\cdot},{\cdot})$ and $\ip{{\cdot}}{{\cdot}}$ would agree in the sense of the next lemma. To relate the two Hermitian forms we consider the $\R$-linear map $\Hprim^4(V;\R)\to H^4_\w(V;\C)$ which is the inclusion $$ H^4_0(V;\R)\to H^4(V;\R)\to H^4(V;\C) $$ followed by projection to $\sigma$'s $\w$-eigenspace. This is an isomorphism of real vector spaces. Since the $\Z$-lattice underlying $\Lambda(V)$ is $\Hprim^4(V;\Z)\sset\Hprim^4(V;\R)$, we get a map $Z:\Lambda(V)\tensor_\Z\R\to H^4_\w(V;\C)$. Since the complex structure on $\Lambda(V)\tensor_\Z\R$ is defined by taking $\w$ to act as $\sigma^*$, and since $H^4_\w$ is defined as a space on which $\sigma^*$ acts by multiplication by $\w$, $Z$ is complex-linear. Since $\Lambda(V)\tensor_\Z\R=\Lambda(V)\tensor_\E\C$, we may regard $Z$ as an isomorphism $\Lambda(V)\tensor_\E\C\to H^4_\w(V;\C)$ of complex vector spaces. \begin{lemma} \label{lem-Hermitian-forms-agree} For all $\a,\b\in \Lambda(V)$, $\bigl(Z(\a),Z(\b)\bigr)=\ip{\a}{\b}$. \end{lemma} \begin{proof} Since both $({\cdot},{\cdot})$ and $\ip{{\cdot}}{{\cdot}}$ are Hermitian forms, it suffices to check that $(Z\a,Z\a)=\ip{\a}{\a}$ for all $\a$. We write $\a_\w$ and $\a_\wbar$ for the projections of $\a\in\Hprim^4(V;\R)$ to $H^4_\w(V;\C)$ and $H^4_\wbar(V;\C)$. By definition, $$ (Z\a,Z\a)=3\int_V \a_\w\wedge\overline{\a_\w} =3\int_V\a_\w\wedge\a_\wbar\;. $$ We will write $\sigma$ for $\sigma^*$ throughout the proof. Since $\sigma\a=\w\a_\w+\wbar\a_\wbar$ and $\sigma^{-1}\a=\wbar\a_\w+\w\a_\wbar$, we deduce $\a_\w=-\frac{1}{\theta}(\w\sigma\a-\wbar\sigma^{-1}\a)$ and $\a_\wbar=-\frac{1}{\theta}(-\wbar\sigma\a+\w\sigma^{-1}\a)$. Therefore \begin{equation*} \begin{split} (Z\a,Z\a) % &=3\int \frac{1}{\theta^2}\Bigl[ \w\sigma\a\wedge(-\wbar\sigma\a)+\w\sigma\a\wedge\w\sigma^{-1}\a\\ % &\mathrel{\phantom{=}}\phantom{3\int \frac{1}{\theta^2}\Bigl[} -\wbar\sigma^{-1}\a\wedge(-\wbar\sigma\a) -\wbar\sigma^{-1}\a\wedge\w\sigma^{-1}\a\Bigr]\\ % &=-\int\Bigl[ -\sigma\a\wedge\sigma\a+\wbar\sigma\a\wedge\sigma^{-1}\a\\ % &\mathrel{\phantom{=}}\phantom{-\int\Bigl[} +\w\sigma^{-1}\a\wedge\sigma\a-\sigma^{-1}\a\wedge\sigma^{-1}\a\Bigr]\\ % &=-\int\Bigl[-2\a\wedge\a-\sigma\a\wedge\sigma^{-1}\a\Bigr]\\ % &=2\a\cdot\a+\int\sigma\a\wedge\sigma^{-1}\a\;. \end{split} \end{equation*} To evaluate the second term, we use \begin{equation} \label{eq-foo} \int \sigma\a\wedge\a+\int\sigma\a\wedge\sigma\a+\int\sigma\a\wedge\sigma^{-1}\a=0\;, \end{equation} which follows from $1+\sigma+\sigma^{-1}=0$. Since $\sigma$ is an isomorphism and $\wedge$ is symmetric, the first and last terms of \eqref{eq-foo} are equal, so we get $$ \int\sigma\a\wedge\sigma^{-1}\a =-\frac{1}{2}\int\sigma\a\wedge\sigma\a =-\frac{1}{2}\a\cdot\a\;. $$ This yields $$ (Z\a,Z\a)=\frac{3}{2}\a\cdot\a=\ip{\a}{\a} $$ as desired. \end{proof} From the lemma it follows that $\Lambda(V)$ has signature $(10,1)$ and that $\ch(H^4_\w(V;\C))$ is naturally identified with $\ch(\Lambda(V)\tensor_\E\C)$. We write $\ch(V)$ for either of these complex hyperbolic spaces. In order to study the variation of the Hodge structure of $V$ we must realize our constructions in local systems over $\forms_0$. To do this we use the fact that the family $\V_0$ over $\forms_0$ gives rise to a sheaf $R^4\pi_*(\Z)$ over $\forms_0$. Recall that this is the sheaf associated to the presheaf $U\mapsto H^4(\pi_\V^{-1}(U);\Z)$. Since $\V_0$ is topologically locally trivial, $R^4\pi_*(\Z)$ is a local system of 23-dimensional $\Z$-lattices isomorphic to $H^4(V;\Z)$. The map $\eta:F\mapsto\eta(V)$, for $F\in\forms_0$, is a section over $\forms_0$, and the subsheaf $(R^4\pi_*(\Z))_0$, whose local sections are the local sections of $R^4\pi_*(\Z)$ orthogonal to $\eta$, is a local system of 22-dimensional $\Z$-lattices isomorphic to $\Hprim^4(V;\Z)$. Since $\sigma$ acts on $\V_0$, it acts on $R^4\pi_*(\Z)$; since it preserves $\eta$ it acts on $(R^4\pi_*(\Z))_0$, giving the sheaf the structure of a local system of $\E$-modules isomorphic to $\Lambda(V)$. We call this local system $\Lambda(\V_0)$. The formula \eqref{eq-defn-of-hermitian-form} endows $\Lambda(\V_0)$ with the structure of a local system of Hermitian $\E$-modules. We write $\ch(\V_0)$ for the corresponding local system of hyperbolic spaces. We can also consider the sheaf $R^4\pi_*(\C)$ over $\forms_0$; it is a local system because it is the complexification of $R^4\pi_*(\Z)$. Now, $\sigma$ acts on $R^4\pi_*(\C)$ and we consider its $\w$-eigensheaf $(R^4\pi_*(\C))_\w$, which is a local system of Hermitian vector spaces isometric to $H^4_\w(V;\C)$, hence of signature $(10,1)$, with a corresponding local system of complex hyperbolic spaces. The map $Z$ identifies the local systems $\Lambda(\V_0)\tensor_\E\C$ and $(R^4\pi_*(\C))_\w$, and therefore identifies the two local systems of complex hyperbolic spaces. Therefore we may regard the inclusion $H^{3,1}(V)\to H^4_\w(V)$ as a defining a section \begin{equation} \label{eq-per-map-as-section} \per:\forms_0\to\ch(\V_0)\;. \end{equation} It is holomorphic since the Hodge filtration varies holomorphically. This is the period map; all our results refer to various formulations of it. Next we obtain a concrete description of the $\E$-lattice $\Lambda(V)$, by investigating the monodromy of $\Lambda(\V_0)$. Fix a basepoint $F\in\forms_0$, let $\Gamma(V)$ be the isometry group of $\Lambda(V)$, and let $\rho$ be the monodromy representation $\pi_1(\forms_0,F)\to\Gamma(V)$. By a meridian around a divisor, such as $\D$, we mean the boundary circle of a small disk transverse to the divisor at a generic point of it, traversed once positively. If $W$ is a complex vector space then a complex reflection of $W$ is a linear transformation that fixes a hyperplane pointwise and has finite order${}>1$. If this order is $2$, $3$ or $6$ then it is called a biflection, triflection or hexaflection. If $W$ has a Hermitian form $\ip{\cdot}{\cdot}$, $r\in W$ has nonzero norm and $\zeta$ is a primitive $n$th root of unity, then the transformation \begin{equation} \label{eq-defn-of-complex-reflection} x\mapsto x-(1-\zeta)\frac{\ip{x}{r}}{\ip{r}{r}}r \end{equation} is a complex reflection of order $n$ and preserves $\ip{{\cdot}}{{\cdot}}$. It fixes $r^\perp$ pointwise and sends $r$ to $\zeta r$; we call it the $\zeta$-reflection in $r$. If $r$ has norm~3 and lies in an $\E$-lattice in which $\theta$ divides all inner products, such as $\Lambda(V)$, then \eqref{eq-defn-of-complex-reflection} shows that $\w$-reflection in $r$ also preserves the lattice. \begin{lemma} \label{lem-meridians-act-by-complex-reflections} The image of a meridian under the monodromy representation $\rho:\pi_1(\forms_0,F)\to\Gamma(V)=\aut \Lambda(V)$ is the $\w$-reflection in an element of $\Lambda(V)$ of norm~$3$. \end{lemma} \begin{proof} The argument is much the same as for lemma~5.4 of \cite{ACT}. Let $D$ be a small disk in $\forms$, meeting $\D$ only at its center, and transversally there. We write $F_0$ for the form at the center of $D$. Suppose without loss of generality that the basepoint $F$ of $\forms_0$ is on $\partial D$, and let $\c$ be the element of $\pi_1(\forms_0,F)$ that traverses $\partial D$ once positively. The essential ingredients of the proof are the following. First, $T_0$ has an $A_1$ singularity, so $V_0$ has an $A_2$ singularity; this means that in suitable local analytic coordinates it is given by $x_1^2+\dots+x_4^2+x_5^3=0$. Second, the vanishing cohomology for this singularity, i.e., the Poincar\'e dual of the kernel of $$ H_4(V;\Z)\to H_4(\V|_D;\Z)\isomorphism H_4(V_0;\Z)\;, $$ is a (positive-definite) copy of the $A_2$ root lattice. (An $A_2$ surface singularity has vanishing cohomology a negative-definite copy of this lattice, and the sign changes when the dimension increases by~2.) Third, following Sebastiani-Thom \cite{sebastiani-thom}, this lattice may be described as $$ V(2)\tensor V(2)\tensor V(2)\tensor V(2)\tensor V(3)\;, $$ where $V(k)$ is the $\Z$-module spanned by the differences of the $k$th roots of unity, $\c$ acts by $$ -1\tensor-1\tensor-1\tensor-1\tensor\w $$ and $\sigma$ acts by $$ 1\tensor1\tensor1\tensor1\tensor\w\;. $$ Here, $\pm1$ (resp.\ $\w$) indicates the action on $V(2)$ (resp.\ $V(3)$) given by sending each square root (resp. cube root) of unity to itself times $\pm1$ (resp.\ $\w$). This shows that the vanishing cohomology is a 1-dimensional $\E$-lattice; we write $r$ for a generator. It also shows that $\c$ acts on $\langle r\rangle$ in the same way that $\sigma^*$ does. Since $\w$'s action on $\Lambda(V)$ is defined to be $\sigma^*$, $\c$ acts on $\langle r\rangle$ by $\w$. Fourth, $\c$ acts trivially on the orthogonal complement of the vanishing cohomology in $H^4(V;\Z)$; this implies that $\c$ is the $\w$-reflection in $r$. Finally, since the roots of the $A_2$ lattice have norm~2, we see by \eqref{eq-defn-of-hermitian-form} that $\ip{r}{r}=3$. \end{proof} The following two lemmas play only a small role in this section, at one point in the proof of theorem~\ref{thm-inner-product-matrix}, to which the reader could skip right away. However, they will be very important in section~\ref{sec-extension}, where we extend the domain of the period map. Their content is that the discriminant has nice local models, which make many homomorphisms from braid groups into $\pi_1(\forms_0)$ visible. We also show that distinct braid generators have distinct monodromy actions. We recall that the fundamental group of the discriminant complement of an $A_n$ singularity is the braid group $B_{n+1}$, also known as the Artin group $\A(A_n)$ of type $A_n$. More generally, the Artin group of an $A_n$, $D_n$ or $E_n$ Dynkin diagram has one generator for each node, with two of the generators braiding ($aba=bab$) or commuting, corresponding to whether the corresponding nodes are joined or not. It is the fundamental group of the discriminant complement of that corresponding singularity \cite{brieskorn-pi1s}. Only $\A(A_n)$ and $\A(D_4)$ will be relevant to this paper. \begin{lemma} \label{lem-discriminant-away-from-E} Suppose $F\in\forms$ defines a cubic threefold with singularities $s_1,\dots,$ $s_m$, each having one of the types $A_n$ or $D_4$, and no other singularities. Let $K_{i=1,\dots,m}$ be the base of a miniversal deformation of a singularity having the type of $s_i$, with discriminant locus $\D_i\sset K_i$. Then there is a neighborhood $U$ of $F$ in $\forms$ diffeomorphic to $K_1\times\dots\times K_m\times B^N$, where $N=35-\sum\dimension K_i$, such that $U-\D$ corresponds to $$ (K_1-\D_1)\times\dots\times(K_m-\D_m)\times B^N\;. $$ In particular, $\pi_1(U-\D)$ is the direct product of $m$ Artin groups, the $i$th factor having the type of the singularity $s_i$. \end{lemma} \begin{proof} This is essentially the assertion that $\forms$ contains a simultaneous versal deformation of all the singularities of $T$. By theorem~1.1 of \cite{du-plessis-wall}, it suffices to show that the sum of the Tjurina numbers of $s_1,\dots,s_m$ is less than~16. Because the singularities of $T$ are quasihomogeneous, their Tjurina numbers coincide with their Milnor numbers. We will write $\mu_i$ for the Milnor number of $T$'s singularity at $s_i$, and $Z_i\sset H_0^4(V)$ for the vanishing cohomology of the corresponding singularity of $V$. If $T$ has a $D_4$ singularity at $s_i$, then $\mu_i=4$, and $V$ has an $\tilde E_6$ singularity there, with $\dimension Z_i=8$ and $\dimension(Z_i\cap Z_i^\perp)=2$. When $T$ has an $A_n$ singularity at $s_i$, we have $\mu_i=n$, and $V$ has a singularity locally modeled on $x_0^2+x_1^2+x_2^2+y^3+z^n=0$. By \cite[p.~77]{arnold-et-al}, $Z_i$ has a basis $a_1,\dots,a_n,b_1,\dots,b_n$ with $a_i^2=b_i^2=2$, $a_i\cdot a_{i\pm1}=-1$, $b_i\cdot b_{i\pm1}=-1$, $a_i\cdot b_i=-1$, $a_i\cdot b_{i-1}=1$, and all other inner products zero. For $n>11$ this quadratic form has a negative-definite subspace of dimension${}\geq4$, so it cannot lie in $H_0^4(V)$, which has signature $(20,2)$. Therefore cubic threefolds cannot have $A_{n>11}$ singularities. For $n<12$, $Z_i$ is nondegenerate except for $n=5$ and~$11$, when $\dimension (Z_i\cap Z_i^\perp)=2$. We made these calculations using PARI/GP \cite{pari}. In every case, we have $\mu_i\leq\frac{2}{3}\dimension\bigl(Z_i/(Z_i\cap Z_i^\perp)\bigr)$. Since $i\neq j$ implies $Z_i\perp Z_j$, we have $$ \sum\dimension \bigl(Z_i/(Z_i\cap Z_i^\perp)\bigr) \leq \dimension H_0^4(V)=22\;. $$ Putting these inequalities together yields $\sum\mu_i\leq\frac{2}{3}\cdot22<16$, so that \cite{du-plessis-wall} applies. This gives the claimed description of $\D$ near $F$, and the description of $\pi_1(U-\D)$ follows immediately. \end{proof} \begin{lemma} \label{lem-local-monodromy-noncyclic} In the situation of the previous lemma, suppose $g$ and $g'$ are two of the standard generators for $\pi_1(U-\D,F')$, where $F'$ is a basepoint in $U-\D$. Then $\rho(g),\rho(g')\in\Gamma(V')$ are the $\w$-reflections in linearly independent roots $r,r'\in \Lambda(V')$. \end{lemma} \begin{proof} If $T$ has two $A_1$ singularities, then near $F$, $\D$ has two components, and $g$ and $g'$ are meridians around them. As we saw in the proof of lemma~\ref{lem-meridians-act-by-complex-reflections}, $V$ has two $A_2$ singularities, and the vanishing cohomology of each of them is a $\sigma$-invariant sublattice of $H_0^4(V')$. In fact, they are the $\E$-spans of $r$ and $r'$. With respect to the cup product, vanishing cocycles for distinct singularities are orthogonal, and it follows from \eqref{eq-defn-of-hermitian-form} that they are also orthogonal under $\ip{}{}$. Therefore $r\perp r'$. Since $r$ and $r'$ have nonzero norm, they must be linearly independent. Now allow $T$ to have $A_n$ and/or $D_4$ singularities, and suppose $g$ and $g'$ commute. Then there exists $F_0\in U$ and a neighborhood $U_0$ of $F_0$ in $U$ with the following properties. $T_0$ has just two singularities, both nodes, $U_0$ contains $F'$, $\pi_1(U_0-\D,F')\isomorphism\Z^2$, and $g,g'\in\pi_1(U-\D,F')$ are represented by loops in $U_0$ that are meridians around the two components of $\D$ at $F_0$. These properties follow from Brieskorn's description \cite{brieskorn-versals} of the versal deformations of simple singularities in terms of the corresponding Coxeter groups. By the previous paragraph, $r$ and $r'$ are linearly independent. The same ideas show that if $g$ and $g'$ braid but do not commute, then there exists $F_0\in U$ and a neighborhood $U_0$ of $F_0$ in $U$ with the following properties. $T_0$ has just one singularity, of type $A_2$, $F'\in U_0$, $\pi_1(U_0-\D,F')\isomorphism B_3$, and $g,g'\in\pi_1(U-\D,F')$ are represented by the standard generators of $B_3$. It is easy to see that the set of cubic threefolds having an $A_2$ singularity and no other singularities is irreducible. Therefore: if there is a single counterexample to the lemma, then $\rho(g)=\rho(g')$ for every pair of generators of $\pi_1(U-\D,F')$, for every $F$ as in the lemma. So it suffices to treat the case that $T$ has a single singularity, of type $A_3$, and $g$ and $g'$ are the first two standard generators of $\pi_1(U-\D,F')\isomorphism B_4$. Adjoining the relation $g=g'$ reduces $B_4$ to $\Z$. If $\rho(g)=\rho(g')$, then $\rho|_H$ factors through $\Z$, which implies that all three generators of $B_4$ have the same $\rho$-image. This is impossible by the previous paragraph, because the first and last generators commute. \end{proof} \begin{theorem} \label{thm-inner-product-matrix} For $F\in\forms_0$, $\Lambda(V)$ is isometric to the $\E$-lattice with inner product matrix \begin{equation} \label{eq-inner-product-matrix} \Lambda:= \begin{pmatrix}3\end{pmatrix} % \oplus \begin{pmatrix} 3&\theta&0&0\\ \thetabar&3&\theta&0\\ 0&\thetabar&3&\theta\\ 0&0&\thetabar&3 \end{pmatrix} % \oplus \begin{pmatrix} 3&\theta&0&0\\ \thetabar&3&\theta&0\\ 0&\thetabar&3&\theta\\ 0&0&\thetabar&3 \end{pmatrix} % \oplus \begin{pmatrix} 0&\theta\\ \thetabar&0 \end{pmatrix} \;. \end{equation} \end{theorem} \begin{remarks} Regarding \eqref{eq-inner-product-matrix} as an $11\times11$ matrix $(\lambda_{ij})$, this means that $\Lambda=\E^{11}$, with $$ \ip{\strut(x_1,\dots,x_{11})}{(y_1,\dots,y_{11})}=\sum_{i,j}\lambda_{ij}x_i\bar y_j\;. $$ The four-dimensional lattice appearing twice among the summands is called $E_8^\E$, because its underlying $\Z$-lattice is a scaled copy of the $E_8$ root lattice. \end{remarks} \begin{proof} By section~2 of \cite{allcock-threefolds}, the cubic threefold $T_0$ defined by $$ F_0=x_2^3+x_0x_3^2+x_1^2x_4-x_0x_2x_4-2x_1x_2x_3+x_4^3 $$ has an $A_{11}$ singularity at $[1{:}0{:}\dots{:}0]\in\cp^4$ and no other singularities. By lemma~\ref{lem-discriminant-away-from-E}, we may choose a neighborhood $U$ of $F_0$ and $F\in U-\D$ such that $\pi_1(U-\D,F)\isomorphism B_{12}$. By lemma~\ref{lem-meridians-act-by-complex-reflections}, the standard generators of $B_{12}$ act on $\Lambda(V)$ by the $\w$-reflections $R_1,\dots,R_{11}$ in pairwise linearly independent roots $r_1,\dots,r_{11}\in \Lambda(V)$. The commutation relations imply $r_i\perp r_j=0$ if $j\neq i\pm1$. By the argument of \cite[\S5]{allcock-inventiones}, the relation $R_iR_{i+1}R_i=R_{i+1}R_iR_{i+1}$ implies that $\bigl|\langle r_i|r_{i+1}\rangle\bigr|=\sqrt3$, so after multiplying some of the $r_i$ by scalars, we may take $\ip{r_i}{r_{i+1}}=\theta$. The rank of the inner product matrix of the $r_i$ is $10$. Therefore, if they were linearly independent then they would span $\Lambda$ up to finite index, and the Hermitian form on $\Lambda$ would be degenerate. It is not, so the span must be only $10$-dimensional. By the argument of \cite[\S5]{allcock-inventiones}, the $r_i$ span a sublattice of $\Lambda(V)$ isometric to the direct sum (call it $\Lambda_{10}$) of the last three summands of \eqref{eq-inner-product-matrix}. In \cite{allcock-inventiones} we used a form of signature $(1,9)$ rather than $(9,1)$; this difference is unimportant. One can check directly that $\theta \Lambda_{10}^*=\Lambda_{10}$; the underlying reason is that the real forms of $E_8^\E$ and $\bigl(\begin{smallmatrix}0&\theta\\\thetabar&0\end{smallmatrix}\bigr)$ are scaled copies of even unimodular $\Z$-lattices. Since $\Lambda(V)\sset\theta \Lambda(V)^*$, $\Lambda_{10}$ is a direct summand of $\Lambda(V)$, so $\Lambda(V)\isomorphism (n)\oplus \Lambda_{10}$ for some $n\in\Z$. We have $n>0$ because $\Lambda(V)$ has signature $(10,1)$. For an $\E$-lattice $M$ we define $M^\Z$ to be the $\Z$-module underlying $M$, equipped with the $\Z$-bilinear pairing $\a\cdot\b=\frac{2}{3}\re\ip{\a}{\b}$. Computation shows that $(n)^\Z$ has inner product matrix $\bigl(\begin{smallmatrix}2n/3&-n/3\\-n/3&2n/3\end{smallmatrix}\bigr)$, $(E_8^\E)^\Z$ is the even unimodular $\Z$-lattice $E_8$, and $\bigl(\begin{smallmatrix}0&\theta\\\thetabar&0\end{smallmatrix}\bigr)\strut^\Z$ is the even unimodular $\Z$-lattice $II_{2,2}=\bigl(\begin{smallmatrix}0&1\\1&0\end{smallmatrix}\bigr)\oplus\bigl(\begin{smallmatrix}0&1\\1&0\end{smallmatrix}\bigr)$. Since $(\Lambda(V))^\Z=H^4_0(V;\Z)$ has determinant~$\pm3$, we must have $n=3$. \end{proof} We define a framing of a form $F\in\forms_0$ to be an equivalence class $[\phi]$ of isometries $\phi:\Lambda(V)\to \Lambda$, two isometries being equivalent if they differ by multiplication by a scalar. Sometimes we write $\phi$ rather than $[\phi]$ and leave it to the reader to check that the construction at hand depends only on $[\phi]$. We define $\framed_0$ to be the set of all framings of all smooth cubic forms. Since the stalk of $\Lambda(\V_0)$ at $F\in\forms_0$ is canonically isomorphic to $\Lambda(V)$, the set $\framed_0$ is in natural bijection with the subsheaf of $P\!\Hom\bigl(\Lambda(\V_0),\forms_0\times \Lambda\bigr)$ consisting of projective equivalence classes of homomorphisms which are isometries on each stalk. This bijection gives $\framed_0$ the structure of a complex manifold. We refer to an element $(F,[\phi])$ of $\framed_0$ as a framed smooth cubic form. We write $\Gamma$ for $\aut \Lambda$ and $P\Gamma$ for $P\!\aut \Lambda$. On $\framed_0$ are defined commuting actions of $\PGamma$ and $G=\GL_5\C/D$, where $D$ is the group $\{I,\w I,\wbar I\}$. An element $\gamma$ of $\PGamma$ acts on the left by $$ \gamma.(F,[\phi])=(F,[\gamma\circ\phi])\;. $$ This action realizes $\PGamma$ as the group of deck transformations of the covering space $\framed_0\to\forms_0$. An element $g$ of $\GL_5\C$ acts on the right by \begin{equation} \label{eq-GL5C-action-on-framed} \bigl(F,[\phi]\bigr).g=\bigl(F\circ g,[\phi\circ g^{-1}{}^*]\bigr)\;. \end{equation} Here, $\GL_5\C$ acts on $\C^5$ on the left, hence acts on $\forms$ on the right by $(F.g)(x)=F(g.x)$, i.e., $F.g=F\circ g$. We extend $\GL_5\C$'s action on $\C^5$ to $\C^6=\C^5\oplus\C$ by the trivial action on the $\C$ summand. This induces a right action on $\V$ by $(F,x).g=(F.g,g^{-1}x)$. That is, $g$ carries the zero-locus of $(F+x_5^3).g$ to the zero-locus of $F+x_5^3$. The $g^{-1}{}^*$ appearing in \eqref{eq-GL5C-action-on-framed} is the inverse of the induced map on cohomology, which respects the $\E$-module structure since $g$ commutes with $\sigma$. The subgroup $D\sset \GL_5\C$ acts trivially on $\framed_0$ because it acts trivially on $\forms_0$ and by scalars on every $\Lambda(V)$. We now introduce the moduli spaces $\moduli_0$ and $\moduli_0^f$ of smooth and framed smooth cubic threefolds. Since $\forms_0\sset\forms_s$, $G$ acts properly on $\forms_0$, with the quotient $\moduli_0=\forms_0/G$ a complex analytic orbifold and a quasiprojective variety. The properness on $\forms_0$ implies properness on $\framed_0$, so $\moduli_0^f=\framed_0/G$ is a complex analytic orbifold and an analytic space. We will see in lemma~\ref{lem-G-acts-freely-and-per-map-has-full-rank} that $\moduli_0^f$ is a complex manifold. Since the $G$-stabilizer of a point $(F,[\phi])$ of $\framed_0$ is a subgroup of the $G$-stabilizer of $F\in\forms_0$, the covering map $\framed_0\to\PGamma\backslash\framed_0=\forms_0$ descends to an orbifold covering map $\moduli_0^f=\framed_0/G\to\forms_0/G=\moduli_0$. Counting dimensions shows that $\moduli_0$ and $\moduli_0^f$ are 10-dimensional. We write $\ch^{10}$ for $\ch(\Lambda\tensor_\E\C)$. Recall that $H^{3,1}(V;\C)$ is a negative line in the Hermitian vector space $H^4_\w(V;\C)$ and that $Z$ is an isometry $\Lambda(V)\tensor_\E\C\to H^4_\w(V;\C)$. We reformulate the period map \eqref{eq-per-map-as-section} as the holomorphic map $\per:\framed_0\to\ch^{10}$ given by \begin{equation} \label{eq-per-map-formulated-with-framings} g(F,[\phi])=\phi\bigl(Z^{-1}\bigl(H^{3,1}(V;\C)\bigr)\bigr)\;. \end{equation} On the right we have written just $\phi$ for $\phi$'s $\C$-linear extension $\Lambda(V)\tensor_\E\C\to \Lambda\tensor_\E\C$. Since $\ch^{10}$ is a 10-ball and bounded holomorphic functions on $\GL_5\C$ are constant, $\per$ is constant along $\GL_5\C$-orbits, so it descends to a holomorphic map $g:\moduli_0^f\to\ch^{10}$, also called the period map. This map is equivariant with respect to the action of $\PGamma$, so it in turn descends to a map \begin{equation} \label{eq-per-map-as-map-to-ball-quotient} \per:\moduli_0 =\PGamma\backslash\moduli_0^f =\PGamma\backslash\framed_0/\GL_5\C \to\PGamma\backslash\ch^{10}\;, \end{equation} again called the period map. \begin{lemma} \label{lem-G-acts-freely-and-per-map-has-full-rank} $G$ acts freely on $\framed_0$, so that $\moduli_0^f$ is a complex manifold, not just an orbifold. The period map $g:\moduli_0^f\to\ch^{10}$ has rank~$10$ at every point of $\moduli_0^f$. \end{lemma} \begin{proof} We prove the second assertion first. Let $F\in\forms_0$, let $F'\in\forms$ be different from $F$, and let $\e>0$ be small enough that the disk $D=\set{F+tG}{\hbox{$t\in\C$ and $|t|\leq\e$}}$ lies in $\forms_0$. Writing $F_t$ for $F+tG$, we know from the discussion surrounding \eqref{eq-Omega(A)} that $H^{3,1}(V_t)$ is spanned by the residue of $\Omega/(F_t+x_5^3)^2$. Since $\V$ trivializes over $D$, we may unambiguously translate this class into $H^4(V;\C)$; this gives a map $h:D\to H^4(V;\C)$. For sufficiently small $t$, $h(F_t)$ is the element of $\Hom\bigl(H_4(V;\Z),\C\bigr)$ given by $$ \hbox{(an integral 4-cycle $C$)}\mapsto \int_{\partial N}\frac{\Omega}{(F_t+x_5^3)^2} $$ where $N$ the the part of the boundary of a tubular neighborhood of $V$ in $\cp^5$ that lies over a submanifold of $V$ representing $C$. Therefore we may differentiate with respect to $t$ under the integral sign, so the derivative of $h$ at the center of $D$ is the element of $\Hom\bigl(H_4(V;\Z),\C\bigr)$ given by $$ C\mapsto\int_{\partial N}\frac{\Omega}{(F_t+x_5^3)^3} \cdot(-2)\frac{\partial}{\partial t}(F_t+x_5^3)\Biggr|_{t=0} = -2\int_{\partial N}\frac{\Omega F'(x_0,\dots,x_4)}{(F+x_5^3)^3}\;. $$ This lies in $H^4_\w(V;\C)$, and it lies in $H^{3,1}$ if and only if $F'$ lies in the Jacobian ideal of $F+x_5^3$, i.e., if and only if $F'$ lies in the Jacobian ideal of $F$, i.e., if and only if the pencil $\langle F,F'\rangle$ in $\forms$ is tangent to the $G$-orbit of $F$. Upon choosing a framing $\phi$ for $F$ and lifting $D$ to a disk $\tilde D=\{(F_t,[\phi_t])\}$ in $\framed_0$ passing through $(F,[\phi])$, it follows that the derivative of $g:\framed_0\to\ch^{10}$ along $\tilde D$ at $(F,[\phi])$ is zero if and only if $\tilde D$ is tangent to the $G$-orbit of $(F,[\phi])$. Since the orbit has codimension~10, $g$ has rank~$10$. To prove the first assertion, recall that an orbifold chart about the image of $(F,[\phi])$ in $\moduli_0^f$ is $U\to U/H\sset\moduli_0^f$, where $H$ is the $G$-stabilizer of $(F,[\phi])$ and $U$ is a small $H$-invariant transversal to the the $G$-orbit of $(F,[\phi])$. We have just seen that the composition $U\to U/H\sset\moduli_0^f\to\ch^{10}$ has rank~10 and is hence a diffeomorphism onto its image. It follows that $H=\{1\}$. \end{proof} \begin{theorem} \label{thm-main-theorem-smooth-case} The period map $\per:\moduli_0\to\PGamma\backslash\ch^{10}$ is an isomorphism onto its image. \end{theorem} \begin{proof} We begin by proving that if $F$ and $F'$ are generic elements of $\forms_0$ with the same image under $\per$ then they are $G$-equivalent, i.e., $T$ and $T'$ are projectively equivalent. By hypothesis there exists an isometry $b:\Lambda(T)\to \Lambda(T')$ which carries $Z^{-1}(H^{3,1}(V))\in \Lambda(V)\tensor_\E\C$ to $Z^{-1}(H^{3,1}(V'))\in \Lambda(V')\tensor_\E\C$. Passing to the underlying integer lattices, $b$ is an isometry $\Hprim^4(V;\Z)\to\Hprim^4(V';\Z)$ carrying $H^{3,1}(V;\C)$ to $H^{3,1}(V';\C)$. By complex conjugation it also identifies $H^{1,3}(V;\C)$ with $H^{1,3}(V';\C)$, and by considering the orthogonal complement of $H^{3,1}\oplus H^{1,3}$ we see that it identifies $\Hprim^{2,2}(V;\C)$ with $\Hprim^{2,2}(V';\C)$. That is, it induces an isomorphism of Hodge structures. Next: one of $\pm b$ extends to an isometry $H^4(V;\Z)\to H^4(V';\Z)$ carrying $\eta(V)$ to $\eta(V')$. This follows from some lattice-theoretic considerations: if $L$ is a nondegenerate primitive sublattice of a unimodular lattice $M$, that is, $L=(L\tensor\Q)\cap M$, then the projections of $M$ into $L\tensor\Q$ and $L^\perp\tensor\Q$ define an isomorphism of $L^*/L$ with $(L^\perp)^*/L^\perp$. Here the asterisk denotes the dual lattice. It is easy to check that an isometry of $L$ and an isometry of $L^\perp$ together give an isometry of $M$ if and only if their actions on $L^*/L$ and $(L^\perp)^*/L^\perp$ coincide under this identification. Since $\bigl\langle\eta(V)\bigr\rangle^*/\bigl\langle\eta(V)\bigr\rangle\isomorphism \Z/3$, it follows that exactly one of the isometries $$ \bigl\langle\eta(V)\bigr\rangle\oplus\Hprim^4(V;\Z)\to \bigl\langle\eta(V')\bigr\rangle\oplus\Hprim^4(V';\Z)\;, $$ given on the first summand by $\eta(V)\mapsto\eta(V')$, and on the second by $\pm b$, extends to an isometry $H^4(V;\Z)\to H^4(V';\Z)$. From Claire Voisin's theorem \cite{voisin} we deduce that there is a projective transformation $\b$ carrying $V$ to $V'$. One can check that the variety $S$ of smooth cubic fourfolds admitting a triflection is irreducible, so that one can speak of a generic such fourfold. Furthermore, a generic such fourfold admits only one triflection (and its inverse). Since $V$ and $V'$ admit the triflections $\sigma^{\pm1}$ and are generic points of $S$, $\b$ carries the fixed-point set $T$ of $\sigma$ in $V$ to the fixed-point set $T'$ of $\sigma$ in $V'$. That is, $T$ and $T'$ are projectively equivalent. We have proven that the period map from $\moduli_0$ to $\PGamma\backslash\ch^{10}$ is generically injective, and the previous lemma shows that it is a local isomorphism. It follows that it is an isomorphism onto its image. \end{proof} \section{The discriminant near a chordal cubic} \label{sec-discr-near-chordal-cubic} In the next section we will enlarge the domain of the period map $\forms_0\to\PGamma\backslash\ch^{10}$, in order to obtain a map from a compactification of $\moduli_0$ to the Baily-Borel compactification $\overline{\PGamma\backslash\ch^{10}}$. In order to do this we will need to understand the local structure of the discriminant $\D\sset\forms_0$, at least near the threefolds to which we will extend $g$. In \cite{allcock-threefolds} (see also \cite{yokoyama}), the GIT-stability of cubic threefolds is completely worked out. There is one distinguished type of threefold, which we call a chordal cubic, which is the secant variety of the rational normal quartic curve. Except for the chordal cubics and those cubics that are GIT-equivalent to them, a cubic threefold is semistable if and only if it has singularities only of types $A_1,\dots,A_5$ and $D_4$. At such a threefold the local structure of $\D$ is given by lemma~\ref{lem-discriminant-away-from-E}. The rest of this section addresses the nature of $\D$ near the chordal cubic locus. It turns out (see the remark following theorem~\ref{thm-chordal-maps-onto-divisor}) that the period map $P\forms_0\to\PGamma\backslash\ch^{10}$ does not extend to a regular map $P\forms_{ss}\to\overline{\PGamma\backslash\ch^{10}}$. The problem is that it does not extend to the chordal cubic locus. Therefore it is natural to try to enlarge the domain of the period map not to $P\forms_{ss}$ but rather to $(\widehat{P\forms})_{ss}$, where $\widehat{P\forms}$ is the blowup of $P\forms$ along the closure of the chordal cubic locus. The details concerning the GIT analysis and the extension of the period map appear in section~\ref{sec-extension}; at this point we are only motivating the study of the local structure of the proper transform $\Dhat$ of $\D$ along the exceptional divisor $E$. Recall that we defined $\D$ as a subset of $\forms$, but will also write $\D$ for its image in $P\forms$. If $T\in P\forms$ is a chordal cubic then we write $E_T$ for $\pi^{-1}(T)\sset E$, where $\pi$ is the natural projection $\widehat {P\forms}\to P\forms$. (There are a number of projection maps in this paper, such as $\pi_\T$ and $\pi_\V$ in section~\ref{sec-smoothmoduli}, and some others introduced later. To keep them straight, we will use a subscript to indicate the domain for all of them except this one.) $E_T$ may be described as the set of unordered 12-tuples in the rational normal curve $R_T$ which is the singular locus of $T$. To see this, one counts dimensions to find that the chordal cubic locus has codimension~13 in $P\forms$, so $E_T$ is a copy of $P^{12}$. To identify $E_T$ with the set of unordered 12-tuples in $R_T$, consider a pencil of cubic threefolds degenerating to $T$. The 12-tuple may be obtained as the intersection of $R_T$ with a generic member of the pencil; since $R_T$ has degree~4, this intersection consists of 12 points, counted with multiplicity. (If every member of the pencil vanishes identically on $R_T$ then the pencil is not transverse to the chordal cubic locus.) We will indicate an element of $E_T$ by a pair $(T,\tau)$, where $\tau$ is an unordered 12-tuple in $R_T$. We will describe $\Dhat\sset\widehat{P\forms}$ by proving the following two theorems, which are similar to but weaker than lemma~\ref{lem-discriminant-away-from-E}. The first is weaker because it asserts a homeomorphism with a standard model of the discriminant, rather than a diffeomorphism. The second gives a complex-analytic isomorphism, but refers to a finite cover of (an open set in) $\widehat{P\forms}$, branched over $E$. But we don't know any reason that the homeomorphism is theorem~\ref{thm-discr-along-E-topological-model} couldn't be promoted to a diffeomorphism. \begin{theorem} \label{thm-discr-along-E-topological-model} Suppose $T$ is a chordal cubic and $\tau$ is a $12$-tuple in $R_T$, with $m$ singularities, of types $A_{n_1},\dots,A_{n_m}$, where an $A_n$ singularity means a point of multiplicity $n+1$. Let $K_{i=1,\dots,m}$ be the base of a miniversal deformation of an $A_{n_i}$ singularity, with discriminant locus $\D_i\sset K_i$. Then there is a neighborhood $U$ of $(T,\tau)$ in $\widehat{P\forms}$ homeomorphic to $B^1\times K_1\times\dots\times K_m\times B^N$, where $N=33-\sum\dimension K_i$, such that $E$ corresponds to $$ \{0\}\times K_1\times\dots\times K_m\times B^N $$ and $U-\widehat\D$ to $$ B^1\times(K_1-\D_1)\times\dots\times(K_m-\D_m)\times B^N\;. $$ In particular, $$ \pi_1\bigl(U-(\Dhat\cup E)\bigr) \isomorphism \Z\times B_{n_1+1}\times\dots\times B_{n_m+1}\;, $$ where the $\Z$ factor is generated by a meridian of $E$ and the standard generators of the braid group factors are meridians of $\Dhat$. \end{theorem} \begin{theorem} \label{thm-discr-along-E-analytic-model} Suppose $(T,\tau)$, $m$, $n_1,\dots,n_m$, $K_1,\dots,K_m$, $\D_1,\dots,$ $\D_m$ and $N$ are as in theorem~\ref{thm-discr-along-E-topological-model}. Then there exists a neighborhood $U$ of $(T,\tau)$ in $\widehat{P\forms}$ diffeomorphic to $B^1\times B^{33}$, with $U\cap E$ corresponding to $\{0\}\times B^{33}$, such that the following holds. We write $\pi_{\tilde U}:\tilde U\to U$ for the $6$-fold cover of $U$ branched over $U\cap E$, and $(T,\tau)\sptilde$ for the point $\pi_{\tilde U}^{-1}(T,\tau)$. There is a neighborhood $V$ of $(T,\tau)\sptilde$ in $\tilde U$ diffeomorphic to $B^1\times K_1\times \dots\times K_m\times B^N$, such that \begin{equation} \label{eq-local-model-description-of-E} V\cap\pi_{\tilde U}^{-1}(E)=\{0\}\times K_1\times\dots\times K_m\times B^N \end{equation} and \begin{equation} \label{eq-local-model-of-discriminant-complement-along-E} V-\pi_{\tilde U}^{-1}(\widehat\D)= B^1\times(K_1-\D_1)\times\dots\times(K_m-\D_m)\times B^N\;. \end{equation} \end{theorem} %XX Can we use the same $U$ in both? (Doesn't matter but a natural %question.) XX The rest of this section is devoted to proving theorems~\ref{thm-discr-along-E-topological-model} and~\ref{thm-discr-along-E-analytic-model}. It is rather technical, especially lemma~\ref{lem-rigidity-of-discriminant} and beyond; although these theorems are analogues of lemma~\ref{lem-discriminant-away-from-E}, the proofs are much more complicated. \begin{lemma} \label{lem-Deltahat-intersect-E} Suppose $T$ is a chordal cubic and $(T,\tau)\in E_T$. \renewcommand\theenumi{\roman{enumi}} \begin{enumerate} \item \label{item-Deltahat-intersect-E} $(T,\tau)$ lies in $\Dhat$ if and only if $\tau$ has a multiple point. % \item \label{item-limits-of-Ans} If $\tau$ has a point of multiplicity $n+1$, then $(T,\tau)$ is a limit of points of $\widehat{P\forms}$ representing cubic threefolds with $A_n$ singularities. \end{enumerate} \end{lemma} Nowhere else in the paper do we refer to any result or notation introduced from here to the end of this section. %\XX{Check when done} When we refer to the ``standard chordal cubic'', we mean the one defined by \begin{equation} \label{eq-standard-chordal-cubic} F(x_0,\dots,x_4)=\det \begin{pmatrix} x_0&x_1&x_2\\ x_1&x_2&x_3\\ x_2&x_3&x_4 \end{pmatrix} =0\;, \end{equation} which is the secant variety of the rational normal curve parameterized by \begin{equation} \label{eq-parameterization-of-RNC} s\mapsto[1,s,s^2,s^3,s^4]\qquad(s\in P^1). \end{equation} We write $P$ for the point $[1,0,0,0,0]$. \begin{proof}[Proof of lemma~\ref{lem-Deltahat-intersect-E}] We prove \eqref{item-limits-of-Ans} first. We take $T$ to be the standard chordal cubic, and place the multiple point of $\tau$ at $P\in R_T$. We suppose without loss of generality that $[0,0,0,0,1]\in R_T$ is not one of the points of $\tau$. Observe that \begin{equation} \label{eq-family-of-Gs-for-A11} \begin{split} G_{u_1,\dots,u_{12}}=F+x_4^3 &{} +u_1x_4^2x_3 +u_2x_4x_3^2 +u_3x_3^3 +u_4x_3^2x_2 \\&{} +u_5x_3x_2^2 +u_6x_2^3 +u_7x_2^2x_1 +u_8x_2x_1^2 \\&{} +u_9x_1^3 +u_{10}x_1^2x_0 +u_{11}x_1x_0^2 +u_{12}x_0^3 \end{split} \end{equation} restricts to $R_T$ as the polynomial \begin{equation} \label{eq-restriction-of-Gs-to-rational-normal-curve} s^{12}+u_1s^{11}+u_2s^{10}+\dots+u_{11}s+u_{12}\;, \end{equation} where $R_T$ is parameterized as in \eqref{eq-parameterization-of-RNC}. Since $\tau$ has no point at $s=\infty$, there is a choice of $u_1,\dots,u_{12}$ such that $\tau$ is the limiting direction of the pencil $\langle F,G_{u_1,\dots,u_{12}}\rangle$. Since $\tau$ has a point of multiplicity $n+1$ at $P$, we have $u_{12}=\dots=u_{12-n}=0$ and $u_{11-(n+1)}\neq0$. Then singularity analysis as in \cite[sec.~2]{allcock-threefolds} shows that a generic member of the pencil has an $A_n$ singularity at $P$. This proves \eqref{item-limits-of-Ans}. Now we prove \eqref{item-Deltahat-intersect-E}. If $\tau$ has a multiple point then \eqref{item-limits-of-Ans} shows that $(T,\tau)\in\widehat\D$. If $\tau$ has no multiple points, and $\langle F,F'\rangle$ is a pencil in $P\forms$ with limiting direction $\tau$, then $dF'$ vanishes at no point of $R_T$, because otherwise $\tau$ would have a multiple point. Since $dF'$ is nonvanishing along the locus $dF=0$, there exists $\e>0$ and a neighborhood $W$ of $F'$ in $\forms$ such that $dF+\eta\,dF''$ is nowhere-vanishing, for all $0<|\eta|<\e$ and all $F''\in W$, so $F+\eta F''$ defines a smooth threefold. Therefore $(T,\tau)$ has a neighborhood in $\widehat{P\forms}$ disjoint from $\Dhat-E$, so $(T,\tau)\notin\Dhat$. \end{proof} Because of the action of $PG$, proving theorems~\ref{thm-discr-along-E-topological-model} and~\ref{thm-discr-along-E-analytic-model} reduces to a similar but lower-dimensional problem. Let $T$ be the standard chordal cubic. In our arguments, the 12-tuple consisting of 12 points all concentrated at $P$ will play a special role; we call it $\tau_0$. Let $A$ be the affine $11$-space in $P\forms$ consisting of the $G_{0,u_2,\dots,u_{12}}$ of \eqref{eq-family-of-Gs-for-A11}. Let $B$ be the projective space spanned by $A$ and $T$, and $\Bhat$ be its proper transform. $\Bhat$ is where most of our work will take place. Near $(T,\tau_0)$, $\widehat{P\forms}$ is a product $\Bhat\times B^{22}$, in a sense made precise by the following lemma. In order to state it, we observe that the stabilizer of $(T,\tau_0)$ in $PG$ is $\C\semidirect\C^*$, of codimension~$22$. Therefore there exists a small $B^{22}\sset PG$ transverse to $\C\semidirect\C^*$ at $1\in PG$. \begin{lemma} \label{lem-Bhat-transverse-to-orbit} The map $\Bhat\times B^{22}\to\widehat{P\forms}$ given by $(b,g)\mapsto b.g$ is a local diffeomorphism at $\bigl((T,\tau_0),1\bigr)$. \end{lemma} \begin{proof} We write $Y$ for the $PG$-orbit of $(T,\tau_0)$. The lemma amounts to the transversality of $Y$ and $\Bhat$ in $\widehat{P\forms}$ at $(T,\tau_0)$. Since $B$ is transverse to the chordal cubic locus at $T$, it suffices to prove that $Y\cap E_T$ and $\Bhat\cap E_T$ are transverse in $E_T$ at $(T,\tau_0)$. We use $u_1,\dots,u_{12}$ as coordinates around $(T,\tau_0)$ in $E_T$ as in the proof of lemma~\ref{lem-Deltahat-intersect-E}. Then $\Bhat\cap E_T$ has equation $u_1=0$. $Y\cap E_T$ is the curve consisting of binary 12-tuples $$ (s-\lambda)^{12}=s^{12}+12\lambda s^{11}+\dots+\lambda^{12}\;, $$ which passes through $(T,\tau_0)$ when $\lambda=0$ and is transverse to $\Bhat$ there because the $s^{11}$ coefficient (i.e., the $u_1$-coordinate) is linear in $\lambda$. \end{proof} \begin{remark} It doesn't matter for us, but we note that $\Bhat$ and $Y$ are not transverse everywhere. Since $Y\cap E_T$ is a rational normal curve of degree~12 and $\Bhat\cap E_T$ is a hyperplane in $E_T$, they intersect in 12 points, counted with multiplicity. Besides $(T,\tau_0)$, the only place they intersect is at $\lambda=\infty$, so they make 11th-order contact there. \end{remark} The analogues of theorems~\ref{thm-discr-along-E-topological-model} and~\ref{thm-discr-along-E-analytic-model} in this lower-dimensional setting are the the following. It turns out (see the proof of theorem~\ref{thm-discr-along-E-topological-model}) that restricting attention to $\tau_0$, rather than treating general $\tau$, is sufficient. \begin{theorem} \label{thm-homeomorphism-model-lower-dimensional} There exists a neighborhood $U'$ of $(T,\tau_0)$ in $\Bhat$ which is homeomorphic to $B^1\times(U'\cap E)$, such that $U'\cap E$ corresponds to $\{0\}\times (U'\cap E)$ and $U'\cap\Dhat$ to $B^1\times(U'\cap E\cap\Dhat)$. \end{theorem} \begin{theorem} \label{thm-analytic-model-lower-dimensional} There exists a neighborhood $U'$ of $(T,\tau_0)$ in $\Bhat$ which is diffeomorphic to $B^1\times B^{11}$, with $U'\cap E$ corresponding to $\{0\}\times B^{11}$, such that the following holds. We write $\pi_{\tilde U'}:\tilde U'\to U'$ for the $6$-fold cover of $U'$, branched over $U'\cap E$, and $(T,\tau_0)\sptilde$ for the preimage therein of $(T,\tau_0)$. There is a neighborhood $\tilde V'$ of $(T,\tau_0)\sptilde$ in $\tilde U'$, and a neighborhood $W'$ of $(T,\tau_0)$ in $\Bhat\cap E$, such that $\tilde V'$ is diffeomorphic to $B^1\times W'$, such that $$ \tilde V'\cap\pi_{\tilde U'}^{-1}(E)=\{0\}\times W' $$ and $$ \tilde V'\cap\pi_{\tilde U'}^{-1}(\Dhat)=B^1\times\bigl(W'\cap\Dhat\bigr)\;. $$ \end{theorem} %XX Again, can the $U'$'s be taken to be the same? XX These theorems describe $\Dhat$ in a neighborhood of $(T,\tau_0)$ in $\Bhat\cap E$ in terms of its intersection with $E$. Therefore we need to understand $\Bhat\cap E\cap\Dhat$: \begin{lemma} \label{lem-versal-deformations-inside-E-cap-Bhat} Suppose $(T,\tau)\in\Bhat\cap E$, and that none of the points of $\tau$ is $[0,0,0,0,1]$. Let $m$, $n_1,\dots,n_m$, $K_1,\dots,K_m$ and $\D_1,\dots,\D_m$ be as in theorem~\ref{thm-discr-along-E-topological-model}. Let $N'=11-\sum\dimension K_i$. Then there is a neighborhood $Z$ of $(T,\tau)$ in $\Bhat\cap E$ diffeomorphic to $K_1\times\dots\times K_m\times B^{N'}$, such that $Z-\Dhat$ corresponds to $$ (K_1-\D_1)\times\dots\times(K_m-\D_m)\times B^{N'}\;. $$ \end{lemma} \begin{proof} Using coordinates $u_2,\dots,u_{12}$ around $(T,\tau_0)$ as in the proof of lemma~\ref{lem-Deltahat-intersect-E}, and parameterizing $R_T$ by $s$ as in \eqref{eq-parameterization-of-RNC}, the $\tau$'s treated in this lemma are parameterized by the functions $$ f(s)=s^{12}+u_2s^{10}+\dots+u_{11}s+u_{12}\;, $$ i.e., as the monic polynomials with root sum equal to zero. The lemma amounts to the assertion that any singular function in this family admits a simultaneous versal deformation of all its singularities, within the family. The family of functions \begin{equation} \label{eq-foo-3} (s-s_0)^n+c_2(s-s_0)^{n-2}+\dots+c_{n-1}(s-s_0)+c_{n} \end{equation} provides a versal deformation of $(s-s_0)^n$, with every member of the family having the same root sum, namely $ns_0$. Given $f$ as above, we may take a product of terms like \eqref{eq-foo-3}, one for each singularity of $f$. This obviously provides a simultaneous versal deformation, and the root sum of any member of the family is that of $f$, namely $0$. Therefore the family lies in $\Bhat\cap E$. \end{proof} \begin{proof}[Proof of theorem~\ref{thm-discr-along-E-topological-model}, given theorem~\ref{thm-homeomorphism-model-lower-dimensional};] We first claim that $(T,\tau_0)$ has a neighborhood $Z$ in $\Bhat\cap E$ and a neighborhood $U$ in $\widehat{P\forms}$, such that $U$ is homeomorphic to $B^1\times Z\times B^{22}$, with $U\cap E$ corresponding to $\{0\}\times Z\times B^{22}$ and $U\cap\Dhat$ to $B^1\times(Z\cap\Dhat)\times B^{22}$. To get this, apply theorem~\ref{thm-homeomorphism-model-lower-dimensional} to obtain $U'\sset\Bhat$ with the properties stated there, and set $Z$ equal to $U'\cap E$. By shrinking $U'$ and $B^{22}\sset PG$ if necessary, we may suppose by lemma~\ref{lem-Bhat-transverse-to-orbit} that $U'\times B^{22}\to\widehat{P\forms}$ is a diffeomorphism onto a neighborhood of $(T,\tau_0)$ in $\widehat{P\forms}$, which we take to be $U$. Then $$ U \isomorphism U'\times B^{22} \isomorphism B^1\times(U'\cap E)\times B^{22} = B^1\times Z\times B^{22}\;. $$ Here, the first `$\isomorphism$' is a diffeomorphism and the second is a homeomorphism. Also, $U\cap E$ corresponds to $(U'\cap E)\times B^{22}=\{0\}\times Z\times B^{22}$, and $U\cap\Dhat$ to $(U'\cap\Dhat)\times B^{22}=B^1\times(Z\cap\Dhat)\times B^{22}$. Now we observe that by the nature of the claim, the same conclusions apply when $\tau_0$ is replaced by any $\tau\in Z$. Now we use lemma~\ref{lem-versal-deformations-inside-E-cap-Bhat}, which describes $Z\cap\Dhat$ (possibly after shrinking $Z$ to a smaller neighborhood of $(T,\tau)$, which shrinks $U$). The result is that $U$ is homeomorphic to $$ B^1\times K_1\times\dots\times K_m\times B^{N'}\times B^{22}\;, $$ such that $$ U\cap E=\{0\}\times K_1\times\dots\times K_m\times B^{N'}\times B^{22} $$ and $$ U-\Dhat=B^1\times(K_1-\D_1)\times\dots\times(K_m-\D_m)\times B^{N'}\times B^{22}\;. $$ Therefore theorem~\ref{thm-discr-along-E-topological-model} holds for $(T,\tau)$. Obviously, it also holds for and $(T,\tau')\in E_T$ that is equivalent to some $(T,\tau)\in Z$ under the stabilizer $\PGL(2,\C)$ of $T$ in $PG$. It is easy to see that this accounts for every $\tau'$, so the proof of theorem~\ref{thm-discr-along-E-topological-model} is complete. \end{proof} The proof of theorem~\ref{thm-discr-along-E-analytic-model}, given theorem~\ref{thm-analytic-model-lower-dimensional}, is essentially the same. Therefore it remains only to prove theorems~\ref{thm-homeomorphism-model-lower-dimensional} and~\ref{thm-analytic-model-lower-dimensional}. We will treat theorem~\ref{thm-analytic-model-lower-dimensional} first. \begin{lemma} \label{lem-Bhat-transform-equals-transform-Bhat} In a neighborhood of $(T,\tau_0)$, $\widehat{B\cap\D}=\Bhat\cap\Dhat$. \end{lemma} \begin{proof} By lemma~\ref{lem-Bhat-transverse-to-orbit}, we may choose a neighborhood $U'$ of $(T,\tau_0)$ in $\Bhat$, and shrink $B^{22}\sset PG$ if necessary, so that $U'\times B^{22}\to\widehat{P\forms}$ is a diffeomorphism onto a neighborhood of $(T,\tau_0)$ in $\widehat{P\forms}$. Under this identification, $\Dhat-E$ corresponds to $\bigl((\Dhat-E)\cap U'\bigr)\times B^{22}$. To get $\Bhat\cap\Dhat$, we take the closure and then intersect with $U'\times\{\hbox{point}\}$, and to get $\widehat{B\cap\D}$, we intersect with $U'\times\{\hbox{point}\}$ and then take the closure. Clearly, both give the same result. \end{proof} The following technical lemma is impossible to motivate without seeing its use in the proof of theorem~\ref{thm-analytic-model-lower-dimensional}; the reader should skip it and refer back when needed. \begin{lemma} \label{lem-rigidity-of-discriminant} Suppose $\d(u_2,\dots,u_{12})$ is a quasihomogeneous polynomial of weight $132$, where $\wt(u_i)=i$. Suppose also that the $u_{12}^{11}$ and $u_{12}^{11-i}u_{11}^iu_i$ $(i=11,\dots,2)$ terms of $\d$ have nonzero coefficients. Suppose $g:(\C^{11},0)\to(\C^{11},0)$ is the germ of a diffeomorphism such that $\d\circ g$ has no terms of weight~$<132$. Then $g$ preserves the weight filtration, in the sense that \begin{equation} \label{eq-vs-in-term-of-us} u_i\circ g=c_iu_i+p_i(u_2,\dots,u_{12})+q_i(u_2,\dots,u_{12}) \end{equation} for each $i$, where $c_i$ is a nonzero constant, $p_i$ is quasihomogeneous of weight~$i$ with no linear terms, and $q_i$ is an analytic function whose power series expansion has only terms of weight~$>i$. \end{lemma} \begin{proof} We write $v_i$ for $u_i\circ g$, regarded as a function of $u_2,\dots,u_{12}$. One obtains $(\d\circ g)(u_2,\dots,u_{12})$ by beginning with $\d(u_2,\dots,u_{12})$ and replacing each $u_i$ by $v_i(u_2,\dots,u_{12})$. This leads to a big mess, with the coefficients of $\d\circ g$ depending on those of $\d$ and the $v_i$ in a complicated way. Nevertheless, there are some coefficients of $\d\circ g$ to which only one term of $g$ can contribute, and this will allow us to deduce that various coefficients of the $v_i$ vanish. To be able to speak precisely, we make the following definitions. When we refer to a term or monomial of $\d$ (resp. $v_i$), we mean a monomial whose coefficient in $\d$ (resp. $v_i$) is nonzero. If $m$ is a monomial $u_{i_1}\cdots u_{i_n}$, and $\mu_j(u_2,\dots,u_{12})$ is a monomial of $v_{i_j}$ for each $j$, then we say that $m$ produces the monomial $$ \mu=\mu_1(u_2,\dots,u_{12})\cdots\mu_n(u_2,\dots,u_{12})\;. $$ For example, if $v_{12}=u_{12}+u_2^2$ and $v_{11}=u_{11}+u_2$, then $m=u_{12}^2u_{11}$ produces the monomials $u_{12}^2u_{11}$, $u_{12}u_2^2u_{11}$, $u_2^4u_{11}$, $u_{12}^2u_2$, $u_{12}u_2^3$ and $u_2^5$. When we wish to be more specific, we say that $m$ produces $\mu$ by the substitution $\mu_j$ for each $u_{i_j}$. Continuing the example, we would say that $u_{12}^2u_{11}$ produces $u_{12}u_2^3$ by substituting $u_{12}$ for one factor $u_{12}$ of $m$, $u_2^2$ for the other, and $u_2$ for the factor $u_{11}$. We note that even if a monomial $m$ of $\d$ produces a monomial $\mu$, the coefficient of $\mu$ in $\d\circ g$ (or even in $m\circ g$) may still be zero, because of possible cancellation. We will prove the lemma by proving the following assertions $\A_d$ by increasing induction on $d$, and we will prove each $\A_d$ by proving the assertions $\B_{d,i}$ by decreasing induction on $i$. $\A_0$ is vacuously true. The first two steps in the induction, $\B_{1,12}$ and $\B_{1,11}$, require special treatment. Assertion $\A_d$: No $v_i$ has any term of degree${}\leq d$ and weight${}12$ then $\deg m>\deg\mu$, so $m$ cannot produce $\mu$. If $m$ has degree~12, then $m$ could only produce $\mu$ by a linear substitution for each factor. If $m$ has a factor $u_{12}$, then any monomial that $m$ produces by such a substitution is divisible by $u_{12}$, by $\B_{1,12}$. We have proven that a monomial $m$ of $\d$ that produces $\mu$ has degree~12 and no $u_{12}$ factors. Since the average weight of the factors is $132/12=11$, every factor is $u_{11}$, so $m=u_{11}^{12}$, proving our claim. Since $\d\circ g$ has no term $u_{j<11}^{12}$, $v_{11}$ has no term $u_{j<11}$. This proves $\B_{1,11}$. Having proven $\B_{1,12}$ and $\B_{1,11}$, we observe that $v_{12}$ has a linear term because $g$ is a diffeomorphism, and since $v_{12}$ has no term $u_{11},\dots,u_2$, it does have a term $u_{12}$. Similarly, using $\B_{1,11}$ and the fact that $v_{11}$ and $v_{12}$ have linearly independent linear parts, we see that $v_{11}$ has a $u_{11}$ term. We will use these facts in the rest of the proof. Proof that $\A_d$ and $\B_{d,12},\dots,\B_{d,i+1}$ imply $\B_{d,i}$ (for $i=11,\dots,2$, except for $\B_{1,11}$, treated above): If $v_i$ had a term $t$ of degree $d$ and weight${}12$. The only way that $m$ can produce a term of degree $11+d$ is by substituting a monomial of degree${}d$ for some factor $u_{12}$, then either that term has degree exactly $d+1$ and $u_{12}$ is substituted for each of the other $u_{12}$'s, or else the degree of the resulting monomial is more than $11+d$. In neither of these cases can the result be $\mu$, because $\mu$ has fewer than $10$ factors $u_{12}$ and has degree $11+d$. Finally, suppose $m$ has degree~12. If we substitute a term of degree${}132p$}) \end{equation} for some $p\geq1$. Although it is not essential, we will soon see that $p=1$. The third idea is to use singularity theory to describe $\D\cap A$ in a neighborhood of $G_{0,\dots,0}$, in terms of a different set of local coordinates. One can show that the threefold defined by $G_{0,\dots,0}$ has only one singularity, which lies at $P$ and has type $A_{11}$. Furthermore, the family $A$ of threefolds provides a versal deformation of this singularity. Therefore there is a neighborhood $W$ of $G_{0,\dots,0}$ in $A$ such that $\D\cap W$ is a copy of the standard $A_{11}$ discriminant. That is, there are analytic coordinates $v_2,\dots,v_{12}$ on $W$, centered at $G_{0,\dots,0}$, such that $\D\cap W$ is defined by $\d(v_2,\dots,v_{12})=0$. This tells us that $p=1$, because $\d'(u_2,\dots,u_{12})=0$ and $\d(v_2,\dots,v_{12})=0$ define the same variety, and therefore vanish to the same order at the origin. We suppose without loss of generality that $W$ is the unit polydisk with respect to the $v_i$ coordinate system. The fourth idea is to use a sort of rigidity of the $A_{11}$ discriminant. Briefly: since the variety defined by $\d(u_2,\dots,u_{12})=0$ is close to that defined by $\d(v_2,\dots,v_{12})=0$, the $u_i$ must be close the $v_i$. This relationship between coordinate systems will be crucial later in the proof. To make this idea precise, consider the diffeomorphism-germ $g$ of $W$ at $G_{0,\dots,0}$ given by $u_i\circ g=v_i$. Since the image of the locus $\d(v_2,\dots,v_{12})=0$ is the locus $\d(u_2,\dots,u_{12})=0$, and since the first of these is also the locus $\d'(u_2,\dots,u_{12})=0$, we have $\d\circ g=\d'$. Now, a computer calculation using Maple \cite{maple} shows that the 11 terms of $\d$ specified by the hypothesis of lemma~\ref{lem-rigidity-of-discriminant} are nonzero, and \eqref{eq-lowest-order-terms} implies that $\d'$ has no terms of weight${}<132$. The lemma then implies \begin{equation} \label{eq-v-in-terms-of-u} v_i=c_iu_i+p_i(u_2,\dots,u_{12})+q_i(u_2,\dots,u_{12})\;, \end{equation} where the $c_i$, $p_i$ and $q_i$ have the properties stated there. We will actually need not this but rather its inverse, giving the $u_i$ in terms of the $v_i$. To compute this it suffices to work in the formal power series ring; one writes \eqref{eq-v-in-terms-of-u} as $$ u_i=\frac{1}{c_i}v_i-\frac{1}{c_i}p_i(u_2,\dots,u_{12}) +\frac{1}{c_i}q_i(u_2,\dots,u_{12})\;, $$ substitutes these expressions into themselves, and repeats this process infinitely many times. The result is \begin{equation} \label{eq-the-us-in-terms-of-the-vs} u_i=c_i'v_i+p_i'(v_2,\dots,v_{12})+q_i'(v_2,\dots,v_{12})\;, \end{equation} where the $c_i'$, $p_i'$ and $q_i'$ satisfy the same conditions as the $c_i$, $p_i$ and $q_i$, with respect to the weights $\wt(v_i)=i$. The fifth idea is to combine the quasihomogeneous scalings of the $u_i$ and $v_i$ to define a map $\a$ whose image will be the set $\tilde V'$ whose existence is claimed by the lemma. For every $0<|\lambda|<1$ we define $\rhosublambda:W\to W$ to be the quasihomogeneous scaling with respect to the $v$-coordinates: $$ \rhosublambda(v_2,\dots,v_{12})=(\lambda^2v_2,\dots,\lambda^{12}v_{12})\;. $$ For every $\lambda\in\C^*$, we define $\eta_\lambda:A\to A$ to be the quasihomogeneous scaling with respect the the $u$-coordinates: $$ \eta_\lambda(u_2,\dots,u_{12})=(\lambda^2u_2,\dots,\lambda^{12}u_{12})\;. $$ The $\eta_\lambda$ are related to the 1-parameter group \eqref{eq-1-parameter-group} by \begin{equation} \label{eq-relation-to-1-par-subgroup} \c\circ\b\bigl(\lambda,\eta_{\lambda}(a)\bigr) =\sigma_\lambda^{-1}(a)\;, \end{equation} which can be verified by expressing both sides in terms of the $u$-coordinates and expanding. We define $$ \a:\bigl(B^1-\{0\}\bigr)\times W\to\bigl(B^1-\{0\}\bigr)\times A $$ by $\a(\lambda,w)=(\lambda,\eta_{\lambda^{-1}}\rhosublambda(a))$. It is easy to see that $\a$ is injective. The first property of $\a$ is that the preimage of the discriminant under it has a very simple form, namely \begin{equation} \label{eq-main-property-of-alpha} (\c\circ\b\circ\a)^{-1}(\Dhat)=\bigl(B^1-\{0\}\bigr)\times(\D\cap W)\;. \end{equation} To see this, observe that $$ \c\circ\b\circ\a(\lambda,w) =\c\circ\b\bigl(\lambda,\eta_{\lambda^{-1}}\rhosublambda(w)\bigr) =\sigma_\lambda(\rhosublambda(w))\;, $$ which lies in $\D$ if and only if $\rhosublambda(w)$ does, hence if and only if $w$ does. The second property of $\a$ is that it extends to a holomorphic map $B^1\times W\to\C\times A$. To prove this, it suffices by Riemann extension to show that it has a continuous extension. The key step is to compute $\lim_{\lambda\to0}\a(\lambda,w)$. If $w\in W$, then its $v$-coordinates are $v_2(w),\dots,v_{12}(w)$, and the $v$-coordinates of $\rhosublambda(w)$ are $\lambda^2v_2(w),\dots,$ $\lambda^{12}v_{12}(w)$. Using \eqref{eq-the-us-in-terms-of-the-vs}, the $u$-coordinates of $\rhosublambda(w)$ are \begin{align*} u_i\bigl(\rhosublambda(w)\bigr) ={}& c_i'\lambda^iv_i(w) +p_i'\bigl(\lambda^2v_2(w),\dots,\lambda^{12}v_{12}(w)\bigr)\\ &\qquad +q_i'\bigl(\lambda^2v_2(w),\dots,\lambda^{12}v_{12}(w)\bigr)\\ {}={}& \lambda^i\cdot\Bigl(c_i'v_i(w)+p_i'\bigl(v_2(w),\dots,v_{12}(w)\bigr)\Bigr)\\ &\qquad +\bigl(\hbox{terms of degree${}>i$ in $\lambda$}\bigr)\;. \end{align*} Therefore the $u$-coordinates of $\eta_{\lambda^{-1}}\rhosublambda(w)$ are \begin{align*} u_i\bigl(\eta_{\lambda^{-1}}\rhosublambda(w)\bigr) =\lambda^{-i}u_i\bigl(\rhosublambda(w)\bigr) ={}& c_iv_i(w)+p_i'\bigl(v_2(w),\dots,v_{12}(w)\bigr)\\ &\qquad +\hbox{terms involving $\lambda$}\;. \end{align*} The limit as $\lambda\to0$ obviously exists, and provides the desired extension. The third property of $\a$ is that after shrinking $W$, we may suppose that $\a:B^1\times W\to\C\times A$ is a diffeomorphism onto a neighborhood of $(0,G_{0,\dots,0})$. To see this we use the fact that $$ u_i\bigl(\a(0,w)\bigr)=c_i'v_i(w)+p_i'\bigl(v_2(w),\dots,v_{12}(w)\bigr)\;; $$ since the $c_i'$ are nonzero and the $p_i'$ have no linear terms, we may shrink $W$ so that $w\mapsto\a(0,w)$ is injective. Then $\a:B^1\times W\to\C\times A$ gives a diffeomorphism from $\{0\}\times W$ onto a neighborhood of $(0,G_{0,\dots,0})$ in $\{0\}\times A\sset \tilde U'$. We will call this diffeomorphism $\a_0$. Because $\a:B^1\times W\to\tilde U'$ is injective and $\tilde U'$ is normal, $\a$ is a diffeomorphism onto a neighborhood of $(T,\tau_0)\sptilde$ in $\tilde U'$. We define $\tilde V'$ to be the image of $\a$. Unwinding the definitions gives \begin{equation} \label{eq-foobar-1} \bigl(\pi_{\tilde U'}\circ\a\bigr)^{-1}(E)=\{0\}\times W \end{equation} and \begin{equation} \label{eq-foobar-2} \bigl(\pi_{\tilde U'}\circ\a\bigr)^{-1}(\Dhat)=B^1\times(W\cap\Dhat)\;. \end{equation} This is exactly what we want, except that $W$ is a subset of $A$, not a subset of $\Bhat\cap E$. However, $\a_0$ identifies $W$ with a subset of $\{0\}\times A$, which is identified under $\c\circ\b$ with a subset of $\Bhat\cap E$, which we take to be $W'$. The identification $W\isomorphism W'$ identifies $W\cap\Dhat$ with $W'\cap\Dhat$, so we may replace $W$ by $W'$ in \eqref{eq-foobar-1} and \eqref{eq-foobar-2}. This completes the proof. \end{proof} \begin{proof}[Proof of theorem~\ref{thm-homeomorphism-model-lower-dimensional}] Theorem~\ref{thm-analytic-model-lower-dimensional} gives us an analytic description of the 6-fold branched cover $\tilde V'$ of a neighborhood of $(T,\tau_0)$ in $\Bhat$. The idea is to take the quotient by the deck group $\Z/6$ and see what we get. Recall that $\tilde U'\isomorphism\C\times A$, where $A\isomorphism\C^{11}$, and the deck group is generated by $(\lambda,a)\mapsto(\lambda\zeta,a)$, where $\zeta=e^{\pi i/3}$. We will write $\xi$ for this map. $(T,\tau_0)\sptilde$ is the point $(0,G_{0,\dots,0})\in\tilde U'$. Also, $\a:B^1\times W\to\tilde U'$ is an embedding onto a neighborhood $\tilde V'$ of $(T,\tau_0)\sptilde$, where $W$ is a polydisk around $G_{0,\dots,0}$ in $A$. We can't regard $\xi$ as a self-map of $B^1\times W$, because $\tilde V'$ may not be a $\xi$-invariant subset of $\tilde U'$. However, we can take the intersection of the finitely many translates of $\tilde V'$, and let $Z\sset B^1\times W$ be the $\a$-preimage of this intersection. The action of $\xi$ on $Z$ can be worked out by using the definition of $\a$. The result is $\xi(0,w)=(0,w)$, and $$ \xi(\lambda,w)=\bigl(\lambda\zeta,\rho_\lambda^{-1}\circ\rho_\zeta^{-1}\circ\eta_\zeta\circ\rho_\lambda(w)\bigr) $$ for $\lambda\in B^1-\{0\}$. Now, $\pi_{\tilde U'}\circ\a$ carries $Z/\langle\xi\rangle$ homeomorphically to a neighborhood of $(T,\tau_0)$ in $\Bhat$. $(T,\tau_0)$ corresponds to the image of $(0,G_{0,\dots,0})$, and $\Dhat\cap\Bhat$ to the image of $Z\cap\bigl(B^1\times(W\cap\Dhat)\bigr)$. Therefore our goal is to describe $Z/\langle\xi\rangle$ and the image therein of $Z\cap\bigl(B^1\times(W\cap\Dhat)\bigr)$. To take the quotient $Z/\langle\xi\rangle$, we first observe that the `slice of pie' $$ \Sigma=\set{\lambda\in B^1}{\hbox{$\lambda=0$ or $\Arg \lambda\in[0,\pi/3]$}} $$ is a fundamental domain for $\lambda\mapsto \lambda\zeta$ acting on $B^1$, and the quotient $B^1/(\Z/6)$ is got by gluing one edge $$ E_0=\set{\lambda\in B^1}{\hbox{$\lambda=0$ or $\Arg \lambda=0$}} $$ to the other $$ E_{\pi/3}=\set{\lambda\in B^1}{\hbox{$\lambda=0$ or $\Arg \lambda=\pi/3$}} $$ in the obvious way. Similarly, $Z/\langle\xi\rangle$ is homeomorphic to $\bigl(Z\cap(\Sigma\times W)\bigr)/{\sim}$, where $\sim$ is the equivalence relation that each $(s,w)\in Z\cap(E_0\times W)$ is identified with $\xi(s,w)\in Z\cap(E_{\pi/3}\times W)$. On the other hand, consider $(\Sigma\times W)/{\approx}$, where $\approx$ is the equivalence relation that each $(s,w)\in E_0\times W$ is identified with $(s\zeta,w)$. The key step in the proof is the definition of the following map $$ \Phi:\bigl(Z_0\cap(\Sigma\times W)\bigr)/{\sim} \ \to\ (\Sigma\times W)/{\approx}\;, $$ where $Z_0$ is a suitable neighborhood of $(0,G_{0,\dots,0})$ in $Z$, defined below. The map is $\Phi(\lambda,w)=(\lambda,w)$ for $\lambda\in E_0$, and $$ \Phi(\lambda,w)= \Bigl(\lambda,\ \rho_{(\Arg \lambda)/(\pi/3)}^{-1}\circ (\rho_\lambda^{-1}\rho_\zeta^{-1}\eta^\phantominverse_\zeta\rho^\phantominverse_\lambda) \circ\rho_{(\Arg \lambda)/(\pi/3)} (w)\Bigr) $$ otherwise. The definition of $Z_0$ is essentially the subset of $Z$ on which the formula makes sense. That is, if $(\lambda,w)\in Z_0$ with $\lambda\in\Sigma- E_0$, then $$ \eta^\phantominverse_\zeta\rho^\phantominverse_\lambda\rho^\phantominverse_{(\Arg \lambda)/(\pi/3)}(w) \in \rho^\phantominverse_\lambda\rho^\phantominverse_{(\Arg \lambda)/(\pi/3)}(W)\;. $$ Assuming for a moment the existence of this set $Z_0$, it is easy to complete the proof. The key point is that the restriction of $\Phi$ to $Z_0\cap(E_0\times W)$ is the obvious inclusion into $E_0\times W$, while the $\Phi$-image of $(\lambda,w)\in(E_{\pi/3}\times W)$ is $\xi(\lambda\zeta^{-1},w)$. Furthermore, for any $(\lambda,w)\in Z_0\cap(\Sigma\times W)$, $\Phi(\lambda,w)$ lies in $\Sigma\times(W\cap\Dhat)$ if and only if $(\lambda,w)$ itself does. (This uses the fact that the $\rho_\lambda$ are quasihomogeneous scalings of $W$ that preserve $W\cap\Dhat$.) It follows that $Z/\langle\xi\rangle$ is homeomorphic to a neighborhood of $(0,G_{0,\dots,0})$ in $(\Sigma\times W)/{\approx}$, i.e., in $B^1\times W$, with its intersection with $\Dhat$ being $\bigl(\Sigma\times(\Dhat\times W)\bigr)/{\approx}$, i.e., $B^1\times(\Dhat\times W)$. This implies the theorem. All that remains is to construct $Z_0$. First observe that $Z$ contains $B^1\times\{G_{0,\dots,0}\}\sset B^1\times W$. Therefore there exists a neighborhood $W_0$ of $G_{0,\dots,0}$ in $W$ such that $B^1\times W_0\sset Z$. Since $\xi$ is defined on $B^1\times W_0$, we have $\eta_\zeta\rho_{\lambda'}(w)\in\rho_{\lambda'}(W)$ for all $(\lambda',w)\in B^1\times W_0$. Setting $\lambda'=\lambda\cdot(\Arg \lambda)/(\pi/3)$, we see that $\Phi$ is defined on $B^1\times W_0$. Now we set $Z_0$ equal to the intersection of the $\xi$-translates of $B^1\times W_0$. \end{proof} \section{Extension of the period map} \label{sec-extension} In this section we extend the period map $g:P\forms_0\to\PGamma\backslash\ch^{10}$ defined in section~\ref{sec-smoothmoduli} to a larger domain. It turns out that $g$ does extend to all of $P\forms_s$ but not to $P\forms_{ss}$. Replacing $\PGamma\backslash\ch^{10}$ by its Baily-Borel compactification $\overline{\PGamma\backslash\ch^{10}}$ allows us to extend $g$ to most of $P\forms_{ss}$ but not all. The problem is that it does not extend to the chordal cubic locus. We will see a proof of this in section~\ref{sec-hodge-theory-chordal}, but for now we just refer to this to motivate the blowing-up of the chordal cubic locus in $P\forms$ to obtain $\widehat{P\forms}$, and extending $g$ to a regular map $(\widehat{P\forms})_{ss}\to\overline{\PGamma\backslash\ch^{10}}$. In this section we will construct the extension; we rely heavily on lemma~\ref{lem-discriminant-away-from-E} and theorems~\ref{thm-discr-along-E-topological-model} and~\ref{thm-discr-along-E-analytic-model}, which describe the local structure of the discriminant. Recall that we write $\pi$ for the projection $\widehat{P\forms}\to P\forms$, $E$ for the exceptional divisor, $E_T\isomorphism P^{12}$ for the points of $E$ lying over a chordal cubic $T$, and $(T,\tau)$ for a point in $E_T$, where $\tau$ is an unordered 12-tuple in the rational normal curve $R_T$ of which $T$ is the secant variety. The first thing we need to do is describe $(\widehat{P\forms})_s$ and $(\widehat{P\forms})_{ss}$, using \cite{allcock-threefolds} and \cite{reichstein}. To lighten the notation we will write just $\widehat{P\forms}_s$ and $\widehat{P\forms}_{ss}$. In order to discuss GIT-stability we need to choose a line bundle on $\widehat{P\forms}$. The following lemma shows that this choice doesn't matter very much; it follows directly from the considerations of \cite[sec.~2]{reichstein}. \begin{lemma} \label{lem-choice-of-line-bundle-irrelevant} For large enough $d$, the stable and semistable loci of $\widehat{P\forms}$, with respect to the standard $\SL(5,\C)$-action on \begin{equation} \label{eq-choice-of-line-bundle} \mathcal{O}(-E)\tensor\pi^*\bigl(\mathcal{O}(d)\bigr)\;, \end{equation} are independent of $d$. \qed \end{lemma} %% \begin{proof} %% XX This is probably obvious, but I haven't given it enough thought. I %% just charged ahead with Reichstein's paper. XX %% This amounts to unwinding the notation in \cite[sec.~2]{reichstein}. Namely, if %% $L$ is an ample line bundle on $P\forms\isomorphism P^{34}$ and $r$ is %% such that $L^r$ is very ample, then for all sufficiently large $d_0$, %% $K=\mathcal{O}(-E)\tensor\pi^*L^{rd_0}$ is very ample. Choosing such %% a $d_0$, we then consider $K_d=K\tensor\pi^* L^d$. Reichstein proves %% that the notions of stability and semistability with respect to $K_d$ %% are independent of $d$, for large enough $d$. Taking %% $L=\mathcal{O}(1)$, $r=1$, and then choosing $d_0$ as above, %% we see that stability and semistability with respect to %% $\mathcal{O}(-E)\tensor \pi^*\bigl(\mathcal{O}(d_0+d)\bigr)$ is %% independent of $d$ %% for large $d$. This implies the lemma. %% \end{proof} Our notation $\widehat{P\forms}_s$ and $\widehat{P\forms}_{ss}$ refers to the linearization \eqref{eq-choice-of-line-bundle} for large enough $d$. Reichstein's work allows us to describe these sets explicitly: \begin{theorem} \label{thm-GIT-analysis} Suppose $T$ is a cubic threefold not in the closure of the chordal cubic locus, regarded as an element of $\widehat{P\forms}$. Then \renewcommand\theenumi{\roman{enumi}} \begin{enumerate} \item \label{item-stable-threefolds} $T$ is stable if and only if each singularity of $T$ has type $A_1$, $A_2$, $A_3$ or $A_4$; % \item \label{item-semistable-threefolds} $T$ is semistable if and only if each singularity of $T$ has type $A_1,\dots,A_5$ or $D_4$; % \item \label{item-closed-orbits-of-semistable-threefolds} $T$ is strictly semistable with closed orbit in $\widehat{P\forms}_{ss}$ if and only if $T$ is projectively equivalent to one of the threefolds defined by $$ x_0x_1x_2+x_3^3+x_4^3 $$ or $$ F_{A,B}=Ax_2^3+x_0x_3^2+x_1^2x_4-x_0x_2x_4+Bx_1x_2x_3\;, $$ with $A,B\in\C$ and $4A\neq B^2$. \end{enumerate} Now suppose instead that $T$ is in the closure of the chordal cubic locus but is not a chordal cubic. Then every element of $\pi^{-1}(T)\sset\widehat{P\forms}$ is unstable. Finally, suppose that $T$ is a chordal cubic, and $\tau$ is an unordered $12$-tuple in the rational normal curve $R_T$, so that $(T,\tau)\in E_T\sset\widehat{P\forms}$. Then \begin{enumerate} \setcounter{enumi}{3} \item \label{item-stable-12-tuples} $(T,\tau)$ is stable if and only if $\tau$ has no points of multiplicity${}\geq6$; % \item \label{item-semistable-12-tuples} $(T,\tau)$ is semistable if and only if $\tau$ has no points of multiplicity greater than~$6$; % \item \label{item-closed-orbits-of-semistable-12-tuples} $(T,\tau)$ is strictly semistable with closed orbit in $\widehat{P\forms}_{ss}$ if and only if $\tau$ consists of two distinct points of multiplicity~$6$. \end{enumerate} Finally, the points \eqref{item-closed-orbits-of-semistable-12-tuples} of $\widehat{P\forms}$ lie in the closure of the union of the orbits of the $T_{A,B}$ from \eqref{item-closed-orbits-of-semistable-threefolds}. \end{theorem} \begin{remarks} The first threefold described in \eqref{item-closed-orbits-of-semistable-threefolds} has three $D_4$ singularities, and is the unique such cubic threefold. The 2-parameter family $T_{A,B}$ really describes only a 1-parameter set of orbits, because the projective equivalence class is determined by the ratio $4A/B^2\in\cp^1-\{1\}$. These threefolds have exactly two singularities, both of type $A_5$, except when $A=0$, when there is also an $A_1$ singularity. Every cubic threefold with two $A_5$ singularities is projectively equivalent to one of these. If $4A$ and $B^2$ were allowed to be equal and nonzero, then $F_{A,B}$ would define a chordal cubic. All of these assertions are proven in section~5 of \cite{allcock-threefolds}. \end{remarks} \begin{proof} Throughout the proof, we will write $L\sset P\forms$ for the closure of the chordal cubic locus. By theorems~2.1 and~2.3 of \cite{reichstein}, a point of $\widehat{P\forms}-E$ is unstable as an element of $\widehat{P\forms}$ if and only if either (a) it is unstable as an element of $P\forms$, or (b) it is GIT-equivalent in $P\forms$ to an element of $L$. Referring to the stability of cubic threefolds, given by theorems~1.3 and~1.4 of \cite{allcock-threefolds}, this says that $\widehat{P\forms}_{ss}-E$ is the set of $T$'s having no singularities of types other than $A_1,\dots,A_5$ and $D_4$. This justifies \eqref{item-semistable-threefolds}. The same theorems of \cite{reichstein} say that a point of $\widehat{P\forms}-E$ is stable as an element of $\widehat{P\forms}$ if and only if it is stable as an element of $P\forms$. Referring again to \cite{allcock-threefolds}, this says that $\widehat{P\forms}_s-E$ is the set of $T$ having no singularities of types other than $A_1,\dots,A_4$, justifying \eqref{item-stable-threefolds}. Now we prove \eqref{item-closed-orbits-of-semistable-threefolds}. If $T\in\widehat{P\forms}_{ss}-E$ is not in $\widehat{P\forms}_s$, then it has a singularity of type $A_5$ or $D_4$. Then theorem~1.3(i,ii) of \cite{allcock-threefolds} implies that $T$ is GIT-equivalent in $P\forms$, hence in $\widehat{P\forms}$, to one of the threefolds given in \eqref{item-closed-orbits-of-semistable-threefolds}. Theorem~1.2 of \cite{allcock-threefolds} implies that the threefolds given explicitly in \eqref{item-closed-orbits-of-semistable-threefolds} have closed orbits in $P\forms_{ss}$; since the orbits miss $L$, they are also closed in $\widehat{P\forms}_{ss}$. It follows that these orbits are the only orbits in $\widehat{P\forms}_{ss}-E$ that are strictly semistable and closed in $\widehat{P\forms}_{ss}$. This justifies \eqref{item-closed-orbits-of-semistable-threefolds}. Now suppose $T\in L$. If $T$ is not a chordal cubic then it is unstable by theorem~1.4(i) of \cite{allcock-threefolds}, so every point of $\widehat{P\forms}$ lying over $T$ is unstable by theorem~2.1 of \cite{reichstein}. It remains only to discuss stability of pairs $(T,\tau)$ with $T$ a chordal cubic. Our key tool is theorem~2.4 of \cite{reichstein}. This says that $(T,\tau)$ is unstable if and only if it lies in the proper transform of the set of cubic threefolds that are GIT-equivalent to chordal cubics. So our job is to determine this proper transform. If $\tau$ has a point of multiplicity${}>6$, then by lemma~\ref{lem-Deltahat-intersect-E}\eqref{item-limits-of-Ans} it is a limit of threefolds having $A_{n>5}$ singularities. Since $T$ lies in $P\forms_{ss}$ and $P\forms_{ss}$ is open in $P\forms$, $(T,\tau)$ is a limit of semistable threefolds having $A_{n>5}$ singularities. By theorem~1.3 of \cite{allcock-threefolds}, such threefolds are GIT-equivalent to chordal cubics. Then Reichstein's theorem~2.4 shows that $(T,\tau)$ is unstable. Reichstein's theorem also asserts that $(T,\tau)\in\widehat{P\forms}_{ss}$ is non-stable if and only if it lies in the proper transform of $P\forms_{ss}-P\forms_s$. If $\tau$ has a point of multiplicity~$6$, then lemma~\ref{lem-Deltahat-intersect-E}\eqref{item-limits-of-Ans} shows that $(T,\tau)$ is a limit of semistable threefolds having $A_5$ singularities, so it is not stable. This justifies the `if' parts of \eqref{item-stable-12-tuples} and \eqref{item-semistable-12-tuples}. Now, suppose $\tau$ has no point of multiplicity${}>6$. Since $P\forms_{ss}$ is open, $T$ has a neighborhood $U\sset P\forms$ with every member of $U-L$ having only $A_n$ and $D_4$ singularities. (In fact, $D_4$ singularities can be excluded, but this doesn't matter here.) By lemma~\ref{lem-discriminant-away-from-E}, every member of $U-L$ admits in $P\forms$ a simultaneous versal deformation of all its singularities. If some member of $U-L$ had an $A_{n\geq6}$ singularity, then at some point of $U-L$, $\D$ would be locally modeled on the $A_n$ discriminant (times a ball of the appropriate dimension). On the other hand, it follows from theorem~\ref{thm-discr-along-E-analytic-model} that after shrinking $U$ we may suppose that at every point of $U-L$, $\D$ is locally modeled on \begin{equation} \label{eq-foo3} \bigcup_{i=1}^m K_1\times\dots\times K_{i-1}\times\D_i\times K_{i+1}\times\dots\times K_m\times B^N\;, \end{equation} where the notation is as in lemma~\ref{lem-discriminant-away-from-E}. In particular, the $\D_i\sset K_i$ are copies of the $A_k$ discriminants for various $k$'s that are at most $5$. Since \eqref{eq-foo3} is not a copy of an $A_{n\geq6}$ discriminant, $U$ contains no points with an $A_{n\geq6}$ singularity, hence no points GIT-equivalent to chordal cubics. By theorem~2.4 of \cite{allcock-threefolds}, $(T,\tau)$ is not unstable, which is to say that it is semistable. This proves the `only if' part of \eqref{item-semistable-12-tuples}. The same argument, using the fact that members of $\widehat{P\forms}_{ss}-E$ have $A_5$ or $D_4$ singularities, proves the `only if' part of \eqref{item-stable-12-tuples}. Finally, if $\tau$ has a point of multiplicity~$6$, then $(T,\tau)$'s orbit closure in $E_T\cap\widehat{P\forms}_{ss}$ contains $(T,\tau')$, where $\tau'$ has two points of multiplicity~$6$. This is a classical fact about point-sets in $P^1$. This proves \eqref{item-closed-orbits-of-semistable-12-tuples}. To prove the last claim of the theorem, just observe that the restrictions of the $F_{A,B}$ in \eqref{item-closed-orbits-of-semistable-threefolds} to the singular locus of the standard chordal cubic (defined by $F_{1,-2}$) consists of $[1,0,0,0,0]$ and $[0,0,0,0,1]$, each with multiplicity~$6$. Let $A\to 1$ and $B\to -2$. \end{proof} Now that we know how much to enlarge the domain of $g$, we will construct the extension. This relies on an analysis of the local monodromy group at a point of $\widehat{P\forms}$, by which we mean the following. In section~\ref{sec-smoothmoduli} we considered the local system $\Lambda(\V_0)$ over $\forms_0$ and its associated local system $\ch(\V_0)$ of complex hyperbolic spaces. Now, $\Lambda(\V_0)$ does not descend to a local system on $P\forms_0$, but $\ch(\V_0)$ does, because the scalars $\{I,\w I,\wbar I\}\sset\GL(5,\C)$ act on each $\Lambda(V)$ by scalar multiplication. After fixing a basepoint $F\in\forms_0$, we defined $$ \rho:\pi_1(\forms_0,F)\to\Gamma(V):=\aut \Lambda(V) $$ to be the monodromy of $\Lambda(\V_0)$. Analogously, we define, for $T\in P\forms_0$, \begin{equation} \label{eq-defn-of-projective-monodromy-rep} P\rho:\pi_1(P\forms_0,T)\to\PGamma(V)\sset\isom\bigl(\ch(V)\bigr)\;. \end{equation} Henceforth, all references to monodromy refer to $P\rho$ unless otherwise stated. In the arguments below, we will compute the monodromy of various elements of $\pi_1(P\forms_0)$. For convenience we will perform various monodromy calculations with roots of $\Lambda(V)$, but these could all be rephrased in terms of elements of $P\Gamma(V)$. Now suppose $T_0$ or $(T_0,\tau_0)$ is an element of $\widehat{P\forms}$ and $U$ is a suitable small neighborhood of it; for example, $U$ could be as in lemma~\ref{lem-discriminant-away-from-E} or theorem~\ref{thm-discr-along-E-topological-model}. By the local fundamental group we mean $\pi_1\bigl(U-(\widehat\D\cup E),T\bigr)$, where $T$ is a basepoint. By the local monodromy action we mean the restriction of $P\rho$ to the local fundamental group, and by the local monodromy group we mean the image of this homomorphism in $\PGamma(V)$. We will see that $\widehat{P\forms}_s$ is exactly the subset of $\widehat{P\forms}_{ss}$ where the local monodromy group is finite. In order to establish this, we will need to know the monodromy around a meridian of $E$: \begin{lemma} \label{lem-monodromy-around-E} Suppose $\c$ is a meridian around $E$ in $\widehat{P\forms}$, $T$ is a point of $\c$, and $P\rho(\c)$ is the monodromy action of $\c$ on $\ch(V)$. Then there is a direct sum decomposition $$ \Lambda(V)=\Lambda_1\oplus\Lambda_{10}\;, $$ where $\Lambda_1$ is the span of a norm $3$ vector $s$, $P\rho(\c)$ acts on $\ch(V)$ as a hexaflection in $s$, and $\Lambda_{10}$ is isometric to the sum of the last three summands in \eqref{eq-inner-product-matrix}. \end{lemma} This lemma resembles lemma~\ref{lem-meridians-act-by-complex-reflections}; each shows that a certain monodromy action is a complex reflection in a norm~3 vector of $\Lambda(V)$. But there is an essential difference. We have already defined a root of $\Lambda(V)$ to be any norm~3 vector $r$; we refine the language by calling $r$ nodal or chordal root according to whether $\ip{r}{\Lambda(V)}$ it $\theta\E$ or $3\E$. It is easy to see that every root is either nodal or chordal. Lemma~\ref{lem-monodromy-around-E} asserts that the monodromy of a meridian around $E$ is a hexaflection in a chordal root. Lemma~\ref{lem-meridians-act-by-complex-reflections} asserts that the monodromy of a meridian around $\Dhat$ is a triflection in a root, and a simple argument shows that this root must be nodal. (Namely, by considering a threefold with an $A_2$ singularity, one finds two meridians of $\Dhat$, which by lemma~\ref{lem-local-monodromy-noncyclic} act by the $\w$-reflections in linearly independent roots $r$ and $r'$, and satisfy the braid relation. This relation forces $\bigl|\ip{r}{r'}\bigr|=\sqrt3$, so $\ip{r}{r'}$ is a unit times $\theta$.) The `nodal' and `chordal' language reflects the fact that these monodromy transformations arise by considering a degeneration to a nodal threefold or to a chordal cubic. We caution the reader that while it is true that every nodal (resp. chordal) root of $\Lambda(V)$ comes from a nodal (resp. chordal) degeneration, we have not yet proven it. In theorem~\ref{lem-transitivity-on-nodal/chordal-hyperplanes} we show that $\Gamma$ is transitive on nodal and chordal roots of $\Lambda$. (The proof of theorem~\ref{lem-transitivity-on-nodal/chordal-hyperplanes} is independent of the rest of the paper, so it could be read at this point.) \begin{proof}[Proof of lemma~\ref{lem-monodromy-around-E}:] Let $T_0$ be the standard chordal cubic, and $\tau_0$ a 12-tuple in $R_{T_0}$ concentrated at one point. By theorem~\ref{thm-discr-along-E-topological-model}, the local fundamental group at $(T_0,\tau_0)$ is $\Z\times B_{12}$, where $\c$ is a generator of $\Z$ and we write $a_1,\dots,a_{11}$ for standard generators for the braid group. By lemma~\ref{lem-meridians-act-by-complex-reflections}, the $a_i$ act on $\ch(V)$ as triflections, and the 1-dimensional eigenspaces of (lifts of the $a_i$ to) $\Lambda(V)$ are spanned by vectors $r_i$ of norm~3. We take $\Lambda_{10}$ to be the span of the $r_i$. Following the proof of theorem~\ref{thm-inner-product-matrix} shows that $\Lambda_{10}$ is a copy of the direct sum of the last three summands of \eqref{eq-inner-product-matrix}, that $\Lambda_{10}$ is a summand of $\Lambda(V)$, and that $\Lambda_{10}^\perp$ is spanned by a vector of norm~$3$. We write $s$ for such a vector and $\Lambda_1$ for its span. Since $\c$ commutes with the $a_i$, any lift of $P\rho(\c)$ to $\Lambda(V)$ multiplies each $r_i$ by a scalar. Since $r_i\cdot r_{i+1}\neq0$, it multiplies all the $r_i$ by the same scalar, so that it acts on $\Lambda_{10}$ as that scalar. Therefore $P\rho(\c)$ acts on $\ch(V)$ as a complex reflection in $s$ of order $2$, $3$ or $6$ (or acts trivially). Now we show that $P\rho(\c)$ has order~6. We may find a neighborhood of $E-\widehat\D$ in $\widehat{P\forms}$ which is a disk bundle over $E-\widehat\D$. (We do not need all the fibers to ``have the same radius''---they can shrink as one approaches $\widehat\D$.) We choose a ball $B$ around $T_0$ in the chordal cubic locus, and write $N$ for the restriction of this disk bundle to $\pi^{-1}(B)\sset E$. $N$ is a neighborhood of $E_{T_0}-\Dhat$ in $\widehat{P\forms}-\Dhat$, and we may suppose without loss of generality that $\c$ and $a_1,\dots,a_{11}$ lie in $N-E$. Now, $N-E$ is a punctured-disk bundle over $\pi^{-1}(B)$, which in turn is a punctured $P^{12}$-bundle over $B$. Since $B$ is a ball, $\pi^{-1}(B)\to B$ trivializes topologically, so up to homotopy, $N-E$ is a circle-bundle over $E_{T_0}-\widehat\D$. Now, $\pi_1(E_{T_0}-\widehat\D)$ is the 12-strand {\it spherical\/} braid group $B_{12}(S^2)$, so $\pi_1(N-E)$ is a central extension of $B_{12}(S^2)$ by $\Z=\langle\c\rangle$. Furthermore, the local description of the discriminant shows that the generators $a_1,\dots,a_{11}$ map to the corresponding standard generators for $B_{12}(S^2)$. Now, $w=a_1\cdots a_{10}a_{11}^2a_{10}\cdots a_1\in B_{12}$ represents the braid in which the leftmost strand moves in a large circle around all the other strands. Since this is trivial in $B_{12}(S^2)$, $w$ is homotopic in $N-E$ to a member of the central $\Z$, i.e., to a power of $\c$. One can write out the $r_i\in \Lambda(V)$ explicitly, as in \cite[sec.~5]{allcock-inventiones}, and then matrix multiplication shows that $w$ acts on $\ch(V)$ with order~6. Since $P\rho(\c)$ has order dividing 6, and some power of it has order 6, $P\rho(\c)$ itself has order 6. \end{proof} Now we will extend the domain of $g$, in two steps. We will begin with the map $g:P\framed_0\to\ch^{10}$ obtained from \eqref{eq-per-map-formulated-with-framings}, where $P\framed_0$ is the quotient of $\framed_0$ by the action of $\C^*\sset\GL(5,\C)$ given in \eqref{eq-GL5C-action-on-framed}. We enlarge $P\framed_0$ to a space $P\framed_s$, which is the branched cover of $\widehat{P\forms}_s$ associated to the covering space $P\framed_0\to P\forms_0$. Formally, we define $p:P\framed_s\to \widehat{P\forms}_s$ to be the Fox completion of the composition $P\framed_0\to P\forms_0\to\widehat{P\forms}_s$. That is, a point of $P\framed_s$ lying over a point $T$ of $\widehat{P\forms}_s$ is a function $\a$ which assigns to each neighborhood $W$ of $T$ a connected component $\a(W)$ of $p^{-1}(W\cap P\forms_0)$, in such a way that if $W'\sset W$ then $\a(W')\sset\a(W)$. $P\framed_s$ has a natural topology; for details see \cite{fox}. By the naturality of the Fox completion, the actions of $P\Gamma$ and $PG$ extend to $P\framed_s$. Since $P\framed_s\to\widehat{P\forms}_s$ is branched over $\Dhat\cup E$, it is clear that the local structure of $\Dhat$ and $E$ plays a key role in the nature of $P\framed_s$; by studying it we will show that $P\framed_s$ is a complex manifold. The analysis follows (3.3)--(3.10) of \cite{ACT}, but is more complicated. We first need to assemble some known results about certain complex reflection groups. Coxeter \cite{coxeter-braid-group-quotients} noticed that for $n=1,\dots,$ $4$, if one adjoins to the $(n+1)$-strand braid group the relations that the $n$ standard generators have order~$3$, then one obtains a finite complex reflection group. We call this group $R_n$. One can describe the group concretely by choosing vectors $r_1,\dots,r_n$ that span an $n$-dimensional Euclidean complex vector space $V_n$, such that the $i$th generator acts as $\w$-reflection in $r_i$. One may scale the roots in any convenient manner; we take $r_i^2=3$ and refer to them as roots. Then the braid and commutation relations imply that $\bigl|\langle r_i|r_{i\pm1}\rangle\bigr|=\sqrt3$ and all other inner products vanish. By multiplying $r_2,\dots,r_n$ in turn by scalars, we can take $r_i\cdot r_{i+1}=\theta$ for all $i$. The group generated by the reflections in $r_1,\dots,r_n$ is what we call $R_n$. In each case, $r_1,\dots,r_n$ generate an $\E$-lattice, and it turns out that the reflections in $R_n$ are exactly the triflections in the norm~$3$ vectors of this lattice. We write $\H_n$ for the union of the orthogonal complements of all these vectors. %(For reference, when %$n=1$, $2$, $3$ or %$4$, there are $1$, $4$, $12$ or $40$ hyperplanes.) \begin{theorem} \label{thm-finite-reflection-group-facts} For any $n=1,\dots,4$, the pair $\bigl(V_n/R_n,\H_n/R_n\bigr)$ is diffeomorphic to $(\C^n,\D_{A_n})$, where $\D_{A_n}$ is the standard $A_n$ discriminant. $R_n$ acts freely on $V_n-\H_n$, so $V_n-\H_n\to\C^n-\D_{A_n}$ is a covering map. The subgroup of $B_{n+1}=\pi_1(\C^n-\D_{A_n})$ corresponding to this covering space is the kernel of the homomorphism $B_{n+1}\to R_n$ described above. Finally, $V_n\to\C^n$ is the Fox completion of the composition $$ V_n-\H_n\to\C^n-\D_{A_n}\to\C^n\;. $$ \end{theorem} \begin{proof} That $V_n/R_n\isomorphism\C^n$ is the same as the ring of $R_n$-invariants on $V_n$ being a polynomial ring, which it is by work of Shephard and Todd \cite{shephard-todd}. That $\H_n/R_n$ corresponds to the $A_n$ discriminant is part of the main result of Orlik and Solomon \cite[cor.~2.26]{orlik-solomon}. It is known that any finite complex reflection group acts freely on the complement of the mirrors of its reflections. %The freeness of %$R_n$'s action on $V_n-\H_n$ is due to %Kostant?? XX. The subgroup $H$ of $B_{n+1}$ corresponding to the covering space contains the cubes of the meridians of $\D_{A_n}$, since $R_n$ contains the triflections across the components of $\H_n$. Since modding out $B_{n+1}$ by the cubes of the standard generators yields a copy of $R_n$, the cubes of meridians generate $H$, and $B_{n+1}/H\isomorphism R_n$ under the indicated homomorphism. The claim about the Fox completion is a special case of the following: suppose $G$ is a finite group acting linearly and faithfully on a finite-dimensional real vector space $V$, and contains no (real) reflections. Then, writing $V_0$ for the open subset of $V$ on which $G$ acts freely, $V\to V/G$ is the Fox completion of $V_0\to V_0/G\to V/G$. (One just verifies that $V\to V/G$ satisfies the definition of a completion of $V_0\to V/G$. The absence of real reflections in $G$ is required for $V_0$ to be locally connected in $V$, in Fox's terminology.) \end{proof} Now we can describe the Fox completion $P\framed_s$. First we describe it away from the chordal locus, and then at a point in the chordal locus. \begin{theorem} \label{thm-fox-completion-away-from-E} Suppose $T\in\widehat{P\forms}_s-E$ has $n_i$ singularities of type $A_i$, for each $i=1,\dots,4$. Suppose $\breve T\in P\framed_s$ lies over $T$. Then near $\breve T$, $P\framed_s$ has a complex manifold structure, indeed a unique one for which $P\framed_s\to P\forms_s$ is holomorphic. With respect to this structure, $\breve T$ has a neighborhood in $P\framed_s$ diffeomorphic to $$ (B^1)^{n_1}\times(B^2)^{n_2}\times(B^3)^{n_3}\times(B^4)^{n_4}\times B^N\;, $$ where $N=34-n_1-2n_2-3n_3-4n_4$, such that $P\framed_0$ corresponds to $$ (B^1-\H_1)^{n_1}\times(B^2-\H_2)^{n_2}\times(B^3-\H_3)^{n_3}\times(B^4-\H_4)^{n_4}\times B^N\;. $$ The stabilizer of $\breve T$ in $P\Gamma$ is isomorphic to $G_1^{n_1}\times\dots\times G_4^{n_4}$, acting in the obvious way, and the map to $\widehat{P\forms}_s$ is the quotient by this group action. \end{theorem} \begin{proof} By lemma~\ref{lem-discriminant-away-from-E}, $T$ has a neighborhood $U\sset P\forms$ diffeomorphic to \begin{equation} \label{eq-local-model-of-Fox-completion} (B^1/G_1)^{n_1}\times\dots\times(B^4/G_4)^{n_4}\times B^N\;, \end{equation} such that $U\cap P\forms_0$ corresponds to (by theorem~\ref{thm-finite-reflection-group-facts}) \begin{equation} \label{eq-local-model-of-hyperplane-complement-in-Fox-completion} \bigl((B^1-\H_1)/G_1\bigr)^{n_1}\times\dots\times\bigl((B^4-\H_4)/G_4\bigr)^{n_4}\times B^N\;, \end{equation} and the local fundamental group is $B_2^{n_1}\times\dots\times B_5^{n_4}$. We write $T'$ for a basepoint in $U-\Dhat$, so that we can refer to its associated fourfold $V'$. By lemma~\ref{lem-meridians-act-by-complex-reflections}, any standard generator of any of the braid group factors acts on $\ch(V')$ as the $\w$-reflection in a root $r\in \Lambda(V')$. We write $H\sset\Gamma(V')$ for the group generated by all these reflections. The local monodromy group is by definition the projectivization of $H$. By lemma~\ref{lem-local-monodromy-noncyclic}, distinct generators of the local fundamental group give linearly independent roots. Therefore the discussion before theorem~\ref{thm-finite-reflection-group-facts} shows that $H$ is $R_1^{n_1}\times\dots\times R_4^{n_4}$. Since the $\E$-sublattice spanned by the roots is positive-definite, it has lower dimension than $\Lambda(V')$, so $H$ contains no scalars. Therefore $P\rho\bigl(\pi_1(U-\Dhat)\bigr)$ is a copy of $H$. By theorem~\ref{thm-finite-reflection-group-facts}, the covering space of $U-\Dhat$ associated to the kernel of this monodromy is $$ (B^1-\H_1)^{n_1}\times\dots\times(G^4-\H_4)^{n_4}\times B^N\;, $$ with the deck group being $H$, acting in the obvious way. Furthermore, the Fox completion over $U$ is then $$ (B^1)^{n_1}\times\dots\times(B^4)^{n_4}\times B^N\;, $$ with $\breve T$ being the point at the center. Since \eqref{eq-local-model-of-Fox-completion} is a diffeomorphism, not just a homeomorphism, $(B^1)^{n_1}\times\dots\times(B^4)^{n_4}\times B^N\to U$ is complex analytic when the domain is equipped with the standard complex manifold structure. This gives the Fox completion a complex manifold structure such that $P\framed_s\to P\forms_s$ is holomorphic. A standard argument using Riemann extension shows that this structure is unique. \end{proof} \begin{theorem} \label{thm-fox-completion-over-E} Suppose $(T,\tau)\in E\cap\widehat{P\forms}_s$, where $\tau$ has $n_i$ points of multiplicity $i+1$, for each $i=1,\dots,4$. Suppose $(T,\tau)\spbreve\in P\framed_s$ lies over $(T,\tau)$. Then near $(T,\tau)\spbreve$, $P\framed_s$ has a complex manifold structure, indeed a unique one for which $P\framed_s\to\widehat{P\forms}_s$ is holomorphic. With respect to this structure, $(T,\tau)\spbreve$ has a neighborhood diffeomorphic to $$ B^1\times(B^1)^{n_1}\times\dots\times(B^4)^{n_4}\times B^{N-1}\;, $$ where $N$ is as in theorem~\ref{thm-fox-completion-away-from-E}, such that the preimage of $E$ corresponds to \begin{equation} \label{eq-foo-6} \{0\}\times(B^1)^{n_1}\times\dots\times(B^4)^{n_4}\times B^{N-1} \end{equation} and the preimage of $P\forms_0$ corresponds to $$ \bigl(B^1-\{0\}\bigr)\times\bigl(B^1-\H_1\bigr)^{n_1}\times\dots\times\bigl(B^4-\H_4\bigr)^{n_4}\times B^{N-1}\;. $$ The stabilizer of $(T,\tau)\spbreve$ in $P\Gamma$ is isomorphic to $\Z/6\times G_1^{n_1}\times\dots\times G_4^{n_4}$, with the $G_i$'s acting in the obvious way. The $\Z/6$ acts freely away from \eqref{eq-foo-6}. \end{theorem} \begin{proof} This is much the same as the previous proof. The difference is that we don't have a local analytic description of $\Dhat$ near $E$, only weaker results, theorems~\ref{thm-discr-along-E-topological-model} and~\ref{thm-discr-along-E-analytic-model}. We begin with the local monodromy analysis. Theorem~\ref{thm-discr-along-E-topological-model} provides a neighborhood $U$ of $(T,\tau)$ with $$ \pi_1\bigl(U-(E\cup\Dhat)\bigr)\isomorphism\Z\times (B_2)^{n_1}\times\dots\times (B_5)^{n_4}\;, $$ where a generator for the $\Z$ factor is a meridian $\c$ of $E$, and the standard generators for the braid group factors are meridians of $\Dhat$. As in the previous proof, we write $T'$ for a basepoint in $U-(E\cup\Dhat)$, so we can refer to the associated fourfold $V'$. We write $H$ for the subgroup of $\Gamma(V')$ generated by the reflections in the roots associated to the braid group factors. By lemma~\ref{lem-monodromy-around-E}, $P\rho(\c)$ is a hexaflection of $\ch(V')$, which is the projectivization of a hexaflection $S$ of $\Lambda(V')$ in a chordal root $s$ of $\Lambda(V')$. We write $H'$ for $\langle H,S\rangle$. The local monodromy group $P\rho\bigl(\pi_1\bigl(U-(E\cup\Dhat)\bigr)\bigr)$ is the projectivization of $H'$. Following the previous proof shows that $H\isomorphism G_1^{n_1}\times\dots\times G_4^{n_4}$. We claim that $s$ is orthogonal to all the roots of the braid group factors. To prove this, we use the fact that $S$ commutes with $H$, so that for every nodal root $r$ of a braid group factor, the triflection $R$ in $r$ carries $s$ to a multiple of itself. Therefore, either $s$ is orthogonal to all the $r$'s, or else it is proportional to one of them. The latter is impossible because then $s$ would be both nodal and chordal, which is impossible. Since $s$ is orthogonal to the $r$'s, $H'=\Z/6\times H$. Arguing as in the previous proof, $H'$ contains no scalars, so it maps isomorphically to its projectivization. Continuing as before proves the corollary, with ``diffeomorphic'' replaced by ``homeomorphic''. To prove the existence of the complex manifold structure, we proceed in two steps. First, we take $\tilde U$ to be the 6-fold cover of $U$, branched over $U\cap E$. This clearly has a complex manifold structure such that $\pi_{\tilde U}:\tilde U\to U$ is holomorphic. Writing $(T,\tau)\sptilde$ for the preimage of $(T,\tau)$, theorem~\ref{thm-discr-along-E-analytic-model} provides us with a neighborhood $\tilde V$ of $(T,\tau)\sptilde$ with the properties stated there. The important property is that $\tilde V$ is {\it diffeomorphic\/} to $B^1\times(B^1/R_1)^{n_1}\times\dots\times(B^4/R_4)^{n_4}\times B^{N-1}$, such that $\tilde V-\tilde\D$ corresponds to $$ B^1\times\bigl((B^1-\H_1)/R_1\bigr)\times\dots\times \bigl((B^4-\H_4)/R_4\bigr)\times B^{N-1}\;. $$ Taking the branched cover of $\tilde V$ with deck group $H$ gives the claimed complex manifold model of $P\framed_s$ near $