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Let $R$ be a standard graded algebra over a field. We investigate how thesingularities of $R$ affect its $h$-vector, which is the coefficients of thenumerator of its Hilbert series. The most concrete consequences of our workasserts that if $R$ satisfies Serre's condition $(S_r)$ and have reasonablesingularities (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. The main technical tools are sharp bounds on regularityof certain Ext modules, which can be viewed as Kodaira-type vanishingstatements for Du Bois and $F$-singularities. Many corollaries are deduced, forinstance 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.

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