WEEKLY ACTIVITIES

Time: 9h30 – 11h, Wednesday October 9, 2024

Venue: Room 612, A6, Institute of Mathematics-VAST

Abstract: Let $R=oplus_{mgeq 0}R_m$ be a $d$-dimensional standard graded finitely generated domain over an algebraically closed field $R_0=k$ and $m$ be the unique homogeneous maximal ideal of $R$. Given a homogeneous ideal $I$ in $R$, we are interested in studying the growth of the graded components $left(I^ncolon_R m^{infty}right)_m$ as $m$ and $n$ vary over integers. In general, this is a difficult problem because the saturated Rees algebra $oplus_{ngeq 0}left(I^ncolon_R m^{infty}right)t^n$ can be non-Noetherian.

In this talk, we shall analyze this by introducing a function, $$f^{mathrm{sat}}_{I}(x) = limsup_{ntoinfty}dfrac{dim_kleft(I^ncolon_R m^{infty}right)_{lfloor xnrfloor}}{n^{d-1}/d!},$$ which we call the saturation density function of $I$. We shall show that $f^{mathrm{sat}}_{I}(x)$ exists as a limit for all real numbers $x$ and it is continuous. We shall also prove a Rees-type statement for ideal sheaves by using the equality of saturation density functions. Our proofs will use the theory of volume functions developed by Lazarsfeld and others. If time permits, we shall give some examples of such functions in low dimensions.

This talk will be based on two ongoing joint projects with Sudeshna Roy and Vijaylaxmi Trivedi.

Program of Special Semester on Commutative Algebra