The Albanese mapping for a punctual Hilbert scheme. II. Symmetrized differentials and singularities

Let $f:X \to A$ be the canonical mapping from the irreducible and nonsingular 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 an irreducible and nonsingular variety of dimension $2n$, and the "Hilbert-Chow" morphism ${\sigma _n}:H_X^n \to X(n)$ is a birational map which desingularizes $X(n)$. Let ${f_n}:X(n) \to A$ denote the map induced by $f$ by addition on $A$. This paper studies the singularities of the composite morphism

$${\varphi _n}:H_X^n\mathop \to \limits^{{\sigma _n}} X(n)\mathop \to \limits^{{f_n}} A,$$

which is a natural generalization of the mapping $C(n) \to J$, where $C$ is an irreducible and nonsingular curve and $J$ is its Jacobian. Unlike the latter, however, ${\varphi _n}$ need not be smooth for $n \gg 0$. We prove that ${\varphi _n}$ is smooth for $n \gg 0$ only if $f:X \to A$ is smooth (Theorem 3), and over ${\mathbf{C}}$ we prove the converse (Theorem 4). In case $X = A$ is an abelian surface, we show ${\varphi _n}$ is smooth for $n$ prime to the characteristic (Theorem 5), and give a counterexample to smoothness for all $n$ (Theorem 6). We exhibit a connection (over ${\mathbf{C}}$) between singularities of ${\varphi _n}$ and generalized Weierstrass points of $X$ (Theorem 9).

Our method is as follows: We first show that the singularities of ${\varphi _n}$ are the zeros of certain holomorphic $1$-forms on $H_X^n$ which are the "symmetrizations" of holomorphic $1$-forms on $X$. We then study "symmetrized differentials" and their zeros on $H_X^n$ (Theorems 1,2). Our method works for curves $C$ as well; we give an alternative proof of a result of Mattuck and Mayer [10, p. 226] which shows that the zeros of symmetrized differentials on $C(n)$ represent (for $C$ complete nonsingular) the special divisors of degree $n$ on $C$.