Deformations of the Hilbert scheme of points on a del Pezzo surface

Abstract

The Hilbert scheme of $n$ points in a smooth del Pezzo surface $S$ parameterizes zero-dimensional subschemes with length $n$ on $S$. We construct a flat family of deformations of Hilb$^n S$ which can be conceptually understood as the family of Hilbert schemes of points on a family of noncommutative deformations of $S$. Further we show that each deformed Hilb$^n S$ carries a generically symplectic holomorphic Poisson structure. Moreover, the generic deformation of Hilb$^nS$ has a $(k+2)$-dimensional moduli space, where the del Pezzo surface is the blow up of projective plane at $k$ sufficiently general points; and each of the fibers is of the form that we construct. Our work generalizes results of Nevins-Stafford constructing deformations of the Hilbert scheme of points on the plane, and of Hitchin studying those deformations from the viewpoint of Poisson geometry.