Betti numbers and ideal structure of projective subschemes of almost maximal degree

Abstract: Similar to Castelnouvo-Mumford regularity, reduction number is a measure of the complexity of the structure of an algebra. For a projective subscheme $X$, there is a degree upper bound $$\deg(X)\leq \binom{e+r}{r},$$ where $e$ is the codimension and $r$ is the reduction number of the homogeneous coordinate ring. In this talk, I discuss the maximal case $\deg(X)=\binom{e+r}{r}$ and the almost maximal case $\deg(X)=\binom{e+r}{r}-1$. In these cases, it is possible to explicitly describe certain initial ideal of the defining ideal of $X$ and consequently one obtains an explicit description of the Betti table. I also discuss how componentwise linearity is helpful for computing the Betti tables of projective varieties with an almost maximal degree.

This is a joint work with Sijong Kwak.

Mathematics

Audience: advanced learners

IIT Bombay Virtual Commutative Algebra Seminar

Series comments: Note: To get meeting ID,fill the Google form on the web-site of the series sites.google.com/view/virtual-comm-algebra-seminar/home