# 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 Hilb^{n} *S* be the Hilbert scheme that parameterizes 0-dimensional
subschemes of length *n*. In this paper, we construct a flat family of
deformations of Hilb^{n} *S* which can be conceptually understood as the
Hilbert scheme of deformed non-commutative del Pezzo surfaces. Further we show that each deformed Hilb^{n} *S* carries a generically symplectic holomorphic Poisson structure. Moreover, the generic deformation
of Hilb^{n} *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.

Contents Full-Text PDF