UC Berkeley

Commutative Algebra and Algebraic Geometry Seminar

April 16, 2019

939 Evans Hall

3:45PM: Regularity, singularities and $h$-vector of graded algebras

Hai Long Dao

Let $R$ be a standard graded algebra over a field. We investigate how the singularities of $\Spec R$ or $\Proj R$ affect the $h$-vector of $R$, which is the coefficients of the numerator of its Hilbert series. The most concrete consequence of our work asserts that if $R$ satisfies Serre's condition $(S_r)$ and have reasonable singularities (Du Bois on the punctured spectrum or $F$-pure), then $h_0,\dots, h_r\geq 0$. Furthermore the multiplicity of $R$ is at least $h_0+h_1+\dots +h_{r-1}$. We also prove that equality in many cases forces $R$ to be Cohen-Macaulay. This is joint work with Linquan Ma and Matteo Varbaro.

5:00PM: On the tangent space to the Hilbert scheme of points in P^3

Ritvik Ramkumar

The Hilbert scheme of n points in P^2 is smooth of dimension 2n and the tangent space to any (monomial) ideal admits a nice combinatorial description. On the other hand the Hilbert scheme of n points in P^3 is singular and there is a conjecture on what the monomial ideal with the largest tangent space dimension should be. By extending the combinatorial methods used in P^2, we give a proof of a major portion of the conjecture (in a sense we will describe). Along the way we will strengthen previous bounds on the dimension of the tangent space. This is joint (ongoing) work with Alessio Sammartano.

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