On some modules supported in the Chow variety

TL;DR

This paper surveys the algebraic and geometric properties of Chow varieties, focusing on Cohen-Macaulay modules, defining equations, syzygies, and the growth of Tor groups, highlighting open questions and future research directions.

Contribution

It introduces new perspectives on modules supported on Chow varieties, especially Cohen-Macaulay modules, and explores the structure of Tor groups over Veronese subalgebras.

Findings

Analysis of Cohen-Macaulay modules supported on Chow varieties

Development of methods to compute defining equations and syzygies

Demonstration of polynomial growth of Tor groups over Chow varieties

Abstract

The study of Chow varieties of decomposable forms lies at the confluence of algebraic geometry, commutative algebra, representation theory and combinatorics. There are many open questions about homological properties of Chow varieties and interesting classes of modules supported on them. The goal of this note is to survey some fundamental constructions and properties of these objects, and to propose some new directions of research. Our main focus will be on the study of certain maximal Cohen-Macaulay modules of covariants supported on Chow varieties, and on defining equations and syzygies. We also explain how to assemble Tor groups over Veronese subalgebras into modules over a Chow variety, leading to a result on the polynomial growth of these groups.