Regularity, singularities and $h$-vector of graded algebras

Abstract:

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 coefficient 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 has reasonable singularities (Du Bois on the punctured spectrum or $F$-pure), then $h_0$, …, $h_r\geq 0$. 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. The main technical tools are sharp bounds on regularity of certain $\operatorname {Ext}$ modules, which can be viewed as Kodaira-type vanishing statements for Du Bois and $F$-pure singularities. Many corollaries are deduced, for instance that nice singularities of small codimension must be Cohen-Macaulay. Our results build on and extend previous work by de Fernex-Ein, Eisenbud-Goto, Huneke-Smith, Murai-Terai and others.