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

Abstract

Let f : X → A f:\,X \to A be the canonical mapping from an algebraic surface X to its Albanese variety A, X ( n ) X(n) the n-fold symmetric product of X, and H X n 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 2 n 2n , and the “Hilbert-Chow” morphism σ n : H X n → X ( n ) {\sigma _n}:\,H_X^n \to X(n) is a birational map which desingularizes X ( n ) X(n) . This paper studies the composite morphism \[ φ n : H X n → σ n X ( n ) → f n A , {\varphi _n}:\,H_X^n\xrightarrow {{{\sigma _n}}}X(n)\xrightarrow {{{f_n}}}A , \] where f n {f_n} is obtained from f by addition on A. The main result (Part 1 of the paper) is that for n ≫ 0 n \gg 0 , all the fibers of φ n {\varphi _n} are irreducible and of dimension 2 n − q 2n - q , where q = dim ⁡ A q = \dim A . An interesting special case (Part 2 of the paper) arises when X = A X = A is an abelian surface; in this case we show (for example) that the fibers of φ n {\varphi _n} are nonsingular, provided n is prime to the characteristic.