A study of the lex plus powers conjecture.
Let \(R:=k[x_1, \dots, x_n]\), fix \(n\) integers \(a_1, \dots, a_n\) such that \(1 \leq a_1 \leq \cdots \leq a_n\). A monomial ideal \(I \subseteq R\) is called an \(\{a_1, \dots, a_n\}\)-\textit{lex plus powers ideal} if there exist monomials \(m_1, \dots, m_l \in I\) such that: \(x_1^{a_1}, \dots, x_n^{a_n}, m_1, \dots, m_l\) is a minimal set of generators of \(I\); if \(m\) is a monomial such that \(\deg(m) = \deg(m_j)\) for some \(j \in \{ 1, \dots, l\}\) and \(m > m_j\), then \(m \in I\). Conjectures by Eisenbud and others and by Charalambous and Evans claim that \(\{a_1, \dots, a_n\}\)-lex plus powers ideals have extremal properties among ideals containing a regular sequence of degree \(\{a_1, \dots, a_n\}\). These extremal properties are similar to those of lex-ideals. In more details, the first conjecture (LPPH) claims that, given a degree \(d\) and an ideal \(I\) containing an \(\{a_1, \dots, a_n\}\)-regular sequence, then if there exists an \(\{a_1, \dots, a_n\}\)-lex plus powers ideal \(L\) such that the Hilbert functions of \(L\) and \(I\) agree in degree \(d\), then there exists a \(\{a_1, \dots, a_n\}\)-lex plus powers ideal \(L'\) such that the Hilbert functions of \(L'\) and \(I\) agree in degree \(d\) and the Hilbert function of \(L'\) bounds that of \(I\) in degree \(d+1\). The second conjecture (LPP) claims that, after fixing a Hilbert function \({\mathcal H}\) and a list of degrees \(\{a_1, \dots, a_n\}\), then the graded Betti numbers of the \(\{a_1, \dots, a_n\}\)-lex plus powers ideal attaining \({\mathcal H}\) should be everywhere larger then the graded Betti numbers of any \(I \subseteq R\) such that \(I\) contains an \(\{a_1, \dots, a_n\}\)-regular sequence, and \(H(R/I)= {\mathcal H}\). The first part of the paper contains several equivalent reformulations of the conjectures. For instance it is proved that LPPH is equivalent to the statement that for a fixed Hilbert function, lex plus powers ideals have the most minimal generators in each degree. Successively, many results are proved. Among them, the author shows that the two conjectures are equivalent in dimension \(\leq 3\) and from this it follows that LPP holds in the monomial case and degree \(\leq 3\). Furthermore, in dimension 2 the two conjectures hold.
scientific article; zbMATH DE number 2032649