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The Albanese mapping for a punctual Hilbert scheme. I. Irreducibility of the fibers

Author: Mark E. Huibregtse
Journal: Trans. Amer. Math. Soc. 251 (1979), 267-285
MSC: Primary 14C05; Secondary 14E99, 14K99
MathSciNet review: 531979
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Abstract: Let $ f:\,X \to A$ be the canonical mapping from an algebraic surface X to its Albanese variety A, $ X(n)$ the n-fold symmetric product of X, and $ H_X^n$ the punctual Hilbert scheme parameterizing 0-dimensional closed subschemes of length n on X. The latter is a nonsingular and irreducible variety of dimension $ 2n$, and the ``Hilbert-Chow'' morphism $ {\sigma _n}:\,H_X^n \to X(n)$ is a birational map which desingularizes $ X(n)$.

This paper studies the composite morphism

$\displaystyle {\varphi _n}:\,H_X^n\xrightarrow{{{\sigma _n}}}X(n)\xrightarrow{{{f_n}}}A ,$

where $ {f_n}$ is obtained from f by addition on A. The main result (Part 1 of the paper) is that for $ n \gg 0$, all the fibers of $ {\varphi _n}$ are irreducible and of dimension $ 2n - q$, where $ q = \dim A$. An interesting special case (Part 2 of the paper) arises when $ X = A$ is an abelian surface; in this case we show (for example) that the fibers of $ {\varphi _n}$ are nonsingular, provided n is prime to the characteristic.

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Keywords: Punctual Hilbert scheme, symmetric product, Albanese variety, Albanese mapping, algebraic surface
Article copyright: © Copyright 1979 American Mathematical Society

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