A combinatorial description of finite O-sequences and ACM genera (Q491249)

Let \(C\) be an arithmetically Cohen-Macaulay (aCM for short) projective curve with a given degree \(d\). Then its Hilbert function \(H_C\) is the 2th integral of a finite O-sequence \(\mathrm{h}=(h_0,h_1,\ldots,h_{s-1})\), which is called the \(\mathrm{h}\)-vector of \(C\). Moreover \(d=h_1+\cdots+h_{s-1}\) and the arithmetic genus \(g\) of \(C\) is determined by its \(h\)-vector, namely, \(g=g(h)=\sum_{j=2}^{s-1}(j-1)h_j\). The paper under reviewed defines a directed graph \(\mathcal{G}_d\) and \(\mathcal{G}_d^s\) for each integer \(d>0\) and \(s>0\) as follows: its vertex set \(V(G_d)\) consists of finite O-sequences of degree \(d\) and its edge set consists of all pairs \((\mathrm{h},\mathrm{h}')\in V(G_d)\times V(G_d)\) such that \(\mathrm{h}'=\mathrm{h}+e_j-e_i\) and \(j>i\). Let \(s>0\). Then \(\mathcal{G}_d^s\) is the subgraph of \(\mathcal{G}_d\) induced on the subset of \(V(G_d)\) consists of finite O-sequences of degree \(d\) and of length \(s\). Let \(G_d\) denote the set of genera of aCM curves of degree \(d\) and \(G^s_d\) the set of genera of finite O-sequence of degree \(d\) and length \(s\). The paper devotes to understand \(G_d\) and \(G^s_d\). More precisely, based on the graphic structures of \(\mathcal{G}_d\) and \(\mathcal{G}_d^s\), the paper under reviewed does the following things. {\parindent=6mm \begin{itemize} \item[-] It provides an effective algorithm for searching aCM genera with given constraints on the multiplicity and the length of the O-sequences. In this algorithm, one inputs an integer \(g>0\), then it returns a finite O-sequence \(\mathrm{h}\) such that \(g(\mathrm{h})=g\) with given constraints on the multiplicity and the length if such an O-sequence exists. \item [-] Determine the maximal element and the minimal elements of \(G_d\) and \(G_d^s\). \item [-] It detects the elements in \(\{1,\ldots, \frac{(d-1)(d-2)}{2}\}\setminus G_d\). \item [-] It construct integers \(m_d\) such that \(\{1,2,\ldots,md\}\subseteq G_d\) and present an algorithm in which one inputs a integer \(d>0\) then it returns the list of all possible aCM genera of a curve of degree \(d\). \item [-] It shows how the search algorithm of aCM genera in 1 helps to detect the minimal Castelnuovo-Mumford regularity of a curve with Cohen-Macaulay postulation, given its degree \(d\) and genus \(g\). \end{itemize}}