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

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 \[ {\varphi _n}: H_X^n\xrightarrow {{{\sigma _n}}}X(n)\xrightarrow {{{f_n}}}A ,\] 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.