Stability of syzygy bundles

Abstract:

We show that given integers $N$, $d$ and $n$ such that ${N\ge 2}, (N,d,n)$ $\ne (2,2,5)$, and ${N+1\le n\le \tbinom {d+N}{N}}$, there is a family of $n$ monomials in $K\left [X_0,\ldots ,X_N\right ]$ of degree $d$ such that their syzygy bundle is stable. Case ${N\ge 3}$ was obtained independently by Coandǎ with a different choice of families of monomials.

For ${(N,d,n)=(2,2,5)}$, there are $5$ monomials of degree $2$ in $K\left [X_0,X_1,X_2\right ]$ such that their syzygy bundle is semistable.