## Commutative Algebra and Algebraic Geometry: The Fellowship of the Ring: Regularity, singularities and h-vector of graded algebras

Seminar | April 16 | 3:45-4:45 p.m. | 939 Evans Hall

Hai Long Dao, University of Kansas

Department of Mathematics

Let $R$ be a standard graded algebra over a field. We investigate how the singularities of $\operatorname {Spec} R$ or $\operatorname {Proj} R$ 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.

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