# Moscow Mathematical Journal

Volume 17, Issue 2, April–June 2017  pp. 291–321.

Deformations of the Hilbert Scheme of Points on a del Pezzo Surface

Authors:  Chunyi Li (1)
Author institution:(1) School of Mathematics and Maxwell Institute, University of Edinburgh

Summary:

Let S be a smooth del Pezzo surface over ℂ of degree d and Hilbn S be the Hilbert scheme that parameterizes 0-dimensional subschemes of length n. In this paper, we construct a flat family of deformations of Hilbn S which can be conceptually understood as the Hilbert scheme of deformed non-commutative del Pezzo surfaces. Further we show that each deformed Hilbn S carries a generically symplectic holomorphic Poisson structure. Moreover, the generic deformation of Hilbn S has an (11−d)-dimensional moduli space and each of the fibers is of the form that we construct.

2010 Math. Subj. Class. 14D20, 16E35.

Keywords: Hilbert scheme, exceptional collection, geometric invariant theory, holomorphic Poisson structure.