UC Berkeley

Commutative Algebra and Algebraic Geometry Seminar

April 02, 2019

939 Evans Hall


3:45PM: Quadratic Gorenstein rings and the Koszul property

Michael Stillman

An artinian local ring (R,m) is called Gorenstein if it has a unique minimal ideal. If R is graded, then it is called Koszul if $R/m$ has a linear R-free resolution. Any Koszul algebra is defined by quadratic relations, but the converse is false, and no one knows a finitely computable criterion. Both types of rings occur in many situations in algebraic geometry and commutative algebra, and in many cases, a Gorenstein quadratic algebra coming from geometry is often Koszul (e.g. homogeneous coordinate rings of most canonical curves) In 2001, Conca, Rossi, and Valla asked the question: must a (graded) quadratic Gorenstein algebra of regularity 3 be Koszul? I will talk about techniques for deciding whether a quadratic Gorenstein algebra is Koszul and methods for generating many examples which are not Koszul. We will explain how these methods provide a negative answer to the above question, as well as a complete picture in the case of regularity at least 4. (This is joint work with Hal Schenck and Matt Mastroeni.

5:00PM: Equivariant completions for degenerations of toric varieties

Netanel Friedenberg

After reviewing classical results about existence of completions of varieties, I will talk about a class of degenerations of toric varieties which have a combinatorial classification - normal toric varieties over rank one valuation rings. I will then discuss recent results about the existence of equivariant completions of such degenerations. In particular, I will show a new result about the existence of normal equivariant completions of these degenerations.

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