Special fibers of coordinate sections of Hankel Matrices

TL;DR

This paper studies special fibers of Hankel determinantal ideals, providing explicit equations, analyzing their algebraic properties, and extending previous work on related algebraic structures.

Contribution

It offers explicit descriptions of the defining equations, proves Cohen-Macaulayness, analyzes minimal generators, and explores the Rees algebra of these special fibers.

Findings

The special fibers are Cohen-Macaulay.

Minimal generators' degrees grow with minors.

Rees algebra is of fiber type in one case.

Abstract

We investigate the special fibers associated with certain coordinate sections of Hankel determinantal ideals. We provide explicit descriptions of their defining equations, showing that these equations admit a natural matrix structure. In particular, we prove that they are Cohen-Macaulay and cannot, in general, be minimally generated only by quadrics and cubics. Instead, we show that the degrees of their minimal generators grow with the size of the minors involved. In one case, we also prove that the Rees algebra is of fiber type. Additionally, we compute algebraic invariants of these special fibers. Our results partially build on and extend the work of Ramkumar and Sammartano on $2$-determinantal ideals and answer some of the questions posed by Cunha, Mostafazadehfard, Ramos, and Simis in earlier work.