Regularity, singularities and \(h\)-vector of graded algebras (Q6567133)

Suppose that \(R=\mathbb{K}[x_1, \ldots, x_n]/I\) is a standard graded algebra over a field \(\mathbb{K}\). Also, recall that the Hilbert series of \(R\) is \(H_R(t)=\sum_{i\in \mathbb{N}}(\mathrm{dim}_{\mathbb{K}}R_i)t^i=\frac{h_0+h_1t+\cdots + h_st^s}{(1-t)^d}\). We call the vector of coefficients \((h_0, h_1, \ldots, h_s)\) the \(h\)-vector of \(R\). The main aim of this work is to explore how the singularities of \(\mathrm{Spec}R\) or \(\mathrm{Proj}R\) affect the \(h\)-vector of \(R\) such that the following theorem is the main result of this paper:\N\N{Theorem 1.2.} Let \(R\) be a standard graded algebra over a field and \((h_0, \ldots, h_s)\) the \(h\)-vector of \(R\). Assume \(R\) satisfies \((S_r)\) and either:\N\begin{itemize}\N\item[(1)] char \(\mathbb{K} = 0\) and \(X = \mathrm{Proj} R\) is Du Bois.\N\item[(2)] char \(\mathbb{K} = p > 0\) and \(R\) is \(F\)-pure.\N\end{itemize}\NThen \(h_i \geq 0\) for \(i = 0, \ldots, r.\) Also, \(h_r +h_{r+1} + \cdots + h_s \geq 0\), or equivalently \(R\) has multiplicity at least \(h_0 + h_1 + \cdots + h_{r-1}.\) If furthermore \(R\) has Castelnuovo-Mumford regularity less than \(r\) or \(h_i = 0\) for some \(i \leq r\), then \(R\) is Cohen-Macaulay. \N\NIn particular, the following corollary says that ``nice singularities of small codimension'' should be Cohen-Macaulay.\N\N{Corollary 5.7.} Let \(R=\mathbb{K}[x_1, \ldots, x_n]/I\) be a standard graded algebra over a field \(\mathbb{K}\) of characteristic \(0\) with \(e = \mathrm{ht}I\) and \(d = \mathrm{dim}R.\) Let \(d_1 \geq d_2 \geq \cdots \) be the degree sequence of a minimal set of generators for \(I\). Assume that \(R\) is unmixed, equidimensional, Cohen-Macaulay in codimension \(l\) and \(X = \mathrm{Proj}R\) has only \(MJ\)-\(log\) canonical singularities. If \(e + l \geq d_1 + \cdots + d_e\), then \(R\) is Cohen-Macaulay.\N\N \bigskip The next theorem is a statement on \(h_2\) that can be seen as a mild extension of Theorem 4.2 in [\textit{D. Eisenbud} and \textit{S. Goto}, J. Algebra 88, 89--133 (1984; Zbl 0531.13015)]: \N\N{Theorem 5.9.} Let \(R=S/I=\mathbb{K}[x_1, \ldots, x_n]/I\) be a standard graded algebra over an algebraically closed field \(\mathbb{K}\) with \(e = \mathrm{ht}I\). Suppose that \(\mathrm{Proj}R\) is connected in codimension one and the radical of \(I\) contains no linear forms. Then \(h_2(R) \geq 0\) and \(e(R) \geq 1 + e\). If equality happens in either case, then \(R_{\mathrm{red}} = S/\sqrt{I}\) is Cohen-Macaulay of minimal multiplicity. \N\NThe authors conclude this paper with the following open question:\N\N{Question 6.1.} Assume that char \(\mathbb{K} = 0\) and \(R\) satisfies \((S_r)\) and is Du Bois in codimension \(r-2\). If \((h_0, \ldots, h_s)\) is the \(h\)-vector of \(R\), is it true that \(h_i \geq 0\) whenever \(i = 0,\ldots, r\)? If furthermore \(R\) has Castelnuovo-Mumford regularity less than \(r\), is it true that \(R\) is Cohen-Macaulay?