The Lex-Plus-Powers Conjecture holds for pure powers

The residuals of lex plus powers ideals and the Eisenbud-Green-Harris conjecture

The $n$-type vectors introduced by Geramita, Harima and Shin are in 1-1 correspondence with the Hilbert functions Artinian of lex ideals. Letting $\mathbb{A} =\{a_1,..., a_n\}$ define the degrees of a regular sequence, we construct ${\rm lpp}_{\le\mathbb{A}}$-vectors which are in 1-1 correspondence with the Hilbert functions of certain lex plus powers ideals (depending on $\mathbb{A}$). This construction enables us to show that the residual of a lex plus powers ideal in an appropriate regular sequence is again a lex plus powers ideal. We then use this result to show that the Eisenbud-Green-Har…

On the Lex-plus-powers Conjecture

Let $S$ be a polynomial ring over a field and $I\subseteq S$ a homogeneous ideal containing a regular sequence of forms of degrees $d_1, \ldots, d_c$. In this paper we prove the Lex-plus-powers Conjecture when the field has characteristic 0 for all regular sequences such that $d_i \geq \sum_{j=1}^{i-1} (d_j-1)+1$ for each $i$; that is, we show that the Betti table of $I$ is bounded above by the Betti table of the lex-plus-powers ideal of $I$. As an application, when the characteristic is 0, we obtain bounds for the Betti numbers of any homogeneous ideal containing a regular sequence of known d…

The lex-plus-power inequality for local cohomology modules

We prove an inequality between Hilbert functions of local cohomology modules supported in the homogeneous maximal ideal of standard graded algebras over a field, within the framework of embeddings of posets of Hilbert functions. As a main application, we prove an analogue for local cohomology of Evans' Lex-Plus-Power Conjecture for Betti numbers. This results implies some cases of the classical Lex-Plus-Power Conjecture, namely an inequality between extremal Betti numbers. In particular, for the classes of ideals for which the Eisenbud-Green-Harris Conjecture is currently known, the projective…

Boij-Söderberg Decompositions of Lex-segment Ideals

Boij-Söderberg theory describes the scalar multiples of Betti diagrams of graded modules over a polynomial ring as a linear combination of pure diagrams with positive coefficients. There are a few results that describe Boij-Söderberg decompositions explicitly. In this paper, we focus on the Betti diagrams of lex-segment ideals and describe the Boij-Söderberg decomposition of a lex-segment ideal in terms of Boij-Söderberg decompositions of some other related lex-segment ideals.

对角超曲面的BCM-正则性及混合特征中的plus-pure阈值

BCM-regularity of diagonal hypersurfaces and plus-pure thresholds in mixed characteristic

本文提出了一种计算plus-pure阈值的新方法，该阈值是log canonical阈值和F-纯阈值的混合特征类比。研究了Fermat型超曲面的BCM-正则性必要条件和充分条件，并建立了混合特征中对角超曲面的plus-pure阈值下界。通过引入Yoshikawa提出的分裂阶序列，给出了混合特征(0,2)中超曲面的plus-pure阈值界，并应用该方法对混合特征(0,2)中的BCM-正则对角超曲面进行了分类。

混合特征下某些尖点型奇点的加纯阈值

Plus-pure thresholds of some cusp-like singularities in mixed characteristic

本文研究混合特征下形如p^a + x^b ∈ Z_p [[x]]的奇点的加纯阈值，证明其在多情况下与对应等特征奇点的F-纯阈值一致，并讨论了零散例子。