Stability of syzygy bundles

TL;DR

This paper proves the existence of stable syzygy bundles for a broad class of monomials in polynomial rings, with specific exceptions, advancing understanding of their stability properties in algebraic geometry.

Contribution

It establishes new existence results for stable syzygy bundles associated with monomials, covering cases previously unresolved, and confirms stability for a wide range of parameters.

Findings

Existence of stable syzygy bundles for given parameters

Special case of semistability for (N,d,n)=(2,2,5)

Independent confirmation for N≥3 by Coand
03

Abstract

We show that given integers $N$, $d$ and $n$ such that ${N\ge2}$, ${(N,d,n)\ne(2,2,5)}$, and ${N+1\le n\le\tbinom{d+N}{N}}$, there is a family of $n$ monomials in $K[X_0,\ldots,X_N]$ of degree $d$ such that their syzygy bundle is stable. Case ${N\ge3}$ was obtained independently by Coand\v{a} with a different choice of families of monomials [Coa09]. For ${(N,d,n)=(2,2,5)}$, there are $5$ monomials of degree~$2$ in $K[X_0,X_1,X_2]$ such that their syzygy bundle is semistable.