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Hilbert Schemes of Points on Surfaces

2005; Birkhäuser; Linguagem: Inglês

10.1007/0-8176-4405-9_7

2296-505X

Lothar Göttsche,

Algebraic structures and combinatorial models

AbstractThe 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.KeywordsPrime DivisorHilbert SchemeSymmetric ProductClosed SubschemeSurjective MorphismThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.