Jet Schemes of the Commuting Matrix Pairs Scheme

TL;DR

This paper investigates the geometric properties of jet schemes over the scheme of commuting matrix pairs, establishing reducibility for large matrix sizes and irreducibility for small sizes, thus revealing size-dependent structural behavior.

Contribution

It proves that for all fixed jet order, the jet scheme becomes reducible as matrix size grows, and confirms irreducibility for 3x3 matrices, extending known results.

Findings

Jet schemes are reducible for large matrix sizes.

Jet schemes are irreducible for 2x2 and 3x3 matrices.

Size of matrices determines the reducibility of jet schemes.

Abstract

We show that for all $k\ge 1$, there exists an integer $N(k)$ such that for all $n\ge N(k)$ the $k$-th order jet scheme over the commuting $n\times n$ matrix pairs scheme is reducible. At the other end of the spectrum, it is known that for all $k\ge 1$, the $k$-th order jet scheme over the commuting $2\times 2$ matrices is irreducible: we show that the same holds for $n=3$.