Rigid ideals by deforming quadratic letterplace ideals

TL;DR

This paper analyzes the deformation space of quadratic letterplace ideals for finite posets with rooted tree Hasse diagrams, revealing their unobstructed deformations and explicit rigidity properties.

Contribution

It explicitly computes the deformation space and the rigid ideal defining the full family of deformations for quadratic letterplace ideals of certain posets.

Findings

Deformations are unobstructed.

The rigid ideal J(2,P) is explicitly computed.

In examples, J(2,P) corresponds to well-known determinantal ideals.

Abstract

We compute the deformation space of quadratic letterplace ideals $L(2,P)$ of finite posets $P$ when its Hasse diagram is a rooted tree. These deformations are unobstructed. The deformed family has a polynomial ring as the base ring. The ideal $J(2,P)$ defining the full family of deformations is a rigid ideal and we compute it explicitly. In simple example cases $J(2,P)$ is the ideal of maximal minors of a generic matrix, the Pfaffians of a skew-symmetric matrix, and a ladder determinantal ideal.