Hilbert Schemes of Points on Surfaces

Abstract

The main aim of this chapter is to prove the following result of Kumar-Thomsen. For any nonsingular split surface X, the Hilbert scheme X^[n] (parametrizing length-n subschemes of X) is split as well. Here, as earlier in this book, by split we mean Frobenius split. The proof relies on some results of Fogarty on the geometry of X[n] and a study of the Hilbert-Chow morphism γ : X^[n] → X⁽ⁿ⁾, where X⁽ⁿ⁾ denotes the n-fold symmetric product of X (parametrizing effective 0-cycles of degree n), and γ maps any length-n subscheme to its underlying cycle.