Regularity, singularities and h -vector of graded algebras

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.