2:00 pm Tuesday, April 17, 2007
Algebra Seminar : Configurations of smooth rational curves in $\mathbb&ob;P&cb;^2$ by Henry Schenck (Texas A&M) in RLM 10.174
One of the most famous open problems in the study of hyperplane arrangements is the following question in commutative algebra: Let $\mathcal&ob;A&cb; = \bigcup H_i \subseteq \mathbb&ob;P&cb;^n$ be defined by the vanishing of $Q=\prod L_i$. When is the jacobian ideal $J_Q$ of Cohen-Macaulay? In this talk, I'll give an overview of how this question relates to the topology of the complement $\mathbb&ob;P&cb;^n \setminus \mathcal&ob;A&cb;$, and discuss some standard techniques for proving $J_Q$ is Cohen-Macaulay. Then I'll restrict to $\mathbb&ob;P&cb;2$, but broaden the investigation to include configurations of smooth rational curves $C_i$, such that $&ob;\mathcal C&cb; = \bigcup_&ob;i=1&cb;^n C_i$ has only ordinary singularities. There are surprising differences which arise (for singularities folks, the difference between milnor and tjurina numbers). Nevertheless, we can obtain interesting inductive criteria for Cohen-Macaulayness. I'll also discuss Terao's conjecture (which states that the Cohen-Macaulay property is combinatorially determined), and show that this conjecture is false in the slightly broader setting of smooth rational curves in $\mathbb&ob;P&cb;^2$. The talk will conclude with a discussion of open problems.
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