Francesco Strazzanti:Rigidity properties of Betti numbers and local cohomology modules

Let $I$ be an ideal of $R=K[x_1,\dots,x_n]$, where $K$ is an infinite field. It is possible to associate a lexicographic ideal $I^{\rm lex}$ with the ideal $I$; the Bigatti-Hulett-Pardue theorem says that the graded Betti numbers of $I^{\rm lex}$ are always greater or equal to the graded Betti numbers of $I$. Several authors showed some ``rigid'' behaviours of the Betti numbers of $I$ with respect to those of its lexicographic ideal and its generic initial ideal; for instance Conca, Herzog, and Hibi proved that if $\beta_{ij}(R/I)=\beta_{ij}(R/I^{\rm lex})$ for some $i$ and all $j$, then the same equalities hold for all $k \geq i$ and all $j$. We quickly survey some of these properties and talk about similar results on Hilbert function of local cohomology modules. Among others, in this context we prove a similar statement of the theorem of Conca, Herzog, and Hibi. This is a joint work with E. Sbarra.