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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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

Author: Mark E. Huibregtse
Journal: Trans. Amer. Math. Soc. 274 (1982), 109-140
MSC: Primary 14C05; Secondary 14C25, 14F07, 14J99, 14K99
MathSciNet review: 670923
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Abstract: 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

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

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$.

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Keywords: Punctual Hilbert scheme, symmetric product, Albanese variety, Albanese mapping, symmetrized differential, algebraic surface
Article copyright: © Copyright 1982 American Mathematical Society