Ideals generated by 2-minors, collection of cells and stack polyominoes

TL;DR

This paper investigates ideals generated by 2-minors associated with collections of cells and stack polyominoes, establishing Cohen--Macaulayness, primality, and class group properties, and classifying Gorenstein stack polyominoes.

Contribution

It extends the theory of ideals generated by 2-minors to general cell collections and stack polyominoes, providing new algebraic and geometric insights.

Findings

Cohen--Macaulayness for convex cell collections

Primality for row or column convex cell collections

Classification of Gorenstein stack polyominoes

Abstract

In this paper we study ideals generated by quite general sets of 2-minors of an $m \times n$-matrix of indeterminates. The sets of 2-minors are defined by collections of cells and include 2-sided ladders. For convex collections of cells it is shown that the attached ideal of 2-minors is a Cohen--Macaulay prime ideal. Primality is also shown for collections of cells whose connected components are row or column convex. Finally the class group of the ring attached to a stack polyomino and its canonical class is computed, and a classification of the Gorenstein stack polyominoes is given.