Ein–Lazarsfeld–Mustopa conjecture for the blow-up of a projective space

Abstract

We solve the Ein-Lazarsfeld-Mustopa conjecture for the blow up of a projective space along a linear subspace. More precisely, let $X$ be the blow up of $\mathbb{P}^n$ at a linear subspace and let $L$ be any ample line bundle on $X$. We show that the syzygy bundle $M_L$ defined as the kernel of the evalution map $H^0(X, L) \otimes \mathcal{O}_X \rightarrow L$ is $L$-stable. In the last part of this note we focus on the rigidness of $M_L$ to study the local shape of the moduli space around the point $\left[M_L\right]$.