WEEKLY ACTIVITIES

Time: 14h00 – 15h30, Wednesday October 16, 2024

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

Abstract: Let $R=k[x_1,ldots,x_d]$ be $d$-dimensional standard graded polynomial ring over a field $k$. Let $mathcal{I} = {I_n}_{ngeq 0}$ and $mathcal{J} = {J_n}_{ngeq 0}$ be two (not necessarily Noetherian) filtrations of homogeneous ideals in $R$ with $I_n subseteq J_n$ for each $ngeq 0$. Suppose that there exists an integer $c>0$ such that ${left(I_nright)}_m = {left(J_nright)}_m$ for all $ngeq 0$ and $mgeq cn$. The limit $lim_{ntoinfty}frac{lambda_Rleft(J_n/I_nright)}{n^d}$ exists due to a result of Cutkosky.

In this talk, we shall consider the function $$f_{mathcal{J}/mathcal{I}}(x) = limsup_{ntoinfty}dfrac{dim_kbig({(J_n)}_{lfloor xnrfloor}/{(I_n)}_{lfloor xnrfloor}big)}{n^d},$$ which we call the density function for $mathcal{J}/mathcal{I}$. We shall show that $f_{mathcal{J}/mathcal{I}}(x)$ exists as a limit for all real numbers $x$. We shall also prove that $f_{mathcal{J}/mathcal{I}}$ is a compactly supported continuous function and $$int_{0}^{infty} f_{mathcal{J}/mathcal{I}}(x)dx = lim_{ntoinfty}frac{lambda_Rleft(J_n/I_nright)}{n^d}.$$ Moreover, we shall describe this function as a difference of volumes of slices of appropriate Newton-Okounkov bodies.

This talk will be based on an ongoing joint project with Sudeshna Roy.

Program of Special Semester on Commutative Algebra