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 [M_{L}].