A characteristic free approach to secant varieties of triple Segre products

TL;DR

This paper investigates secant varieties of triple Segre products using combinatorial algebra, rederives known results in a characteristic-free manner, and computes key algebraic invariants like degree and regularity.

Contribution

It provides a characteristic-independent approach to understanding the defining ideals and properties of secant varieties of triple Segre products, extending previous results.

Findings

Reproved and extended results on defining ideals and Cohen-Macaulay property.

Computed degrees of secant varieties.

Provided sharp bounds for Castelnuovo-Mumford regularity.

Abstract

The goal of this short note is to study the secant varieties of the triple Segre product of type (1,a,b) by means of the standard tools of combinatorial commutative algebra. We reprove and extend to arbitrary characteristic results of Landsberg and Weyman regarding the defining ideal and the Cohen-Macaulay property of the secant varieties. Furthermore for these varieties we compute the degree and give a bound for their Castelnuovo-Mumford regularity which is sharp in many cases.