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

Hilbert scheme;deformation theory;del Pezzo surface

Li, Chunyi

关键词: Hilbert scheme;    deformation theory;    del Pezzo surface;   

Others  :  https://www.ideals.illinois.edu/bitstream/handle/2142/49684/Chunyi_Li.pdf?sequence=1&isAllowed=y

美国|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

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.