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
DOI:
https://doi.org/10.1090/S0002-9947-1982-0670923-0
MathSciNet review:
670923
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Abstract | References | Similar Articles | Additional Information
Abstract: Let be the canonical mapping from the irreducible and nonsingular surface
to its Albanese variety
,
the
-fold symmetric product of
, and
the punctual Hilbert scheme parameterizing 0-dimensional closed subschemes of length
on
. The latter is an irreducible and nonsingular variety of dimension
, and the "Hilbert-Chow" morphism
is a birational map which desingularizes
. Let
denote the map induced by
by addition on
. This paper studies the singularities of the composite morphism

















Our method is as follows: We first show that the singularities of are the zeros of certain holomorphic
-forms on
which are the "symmetrizations" of holomorphic
-forms on
. We then study "symmetrized differentials" and their zeros on
(Theorems 1,2). Our method works for curves
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
represent (for
complete nonsingular) the special divisors of degree
on
.
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Additional Information
DOI:
https://doi.org/10.1090/S0002-9947-1982-0670923-0
Keywords:
Punctual Hilbert scheme,
symmetric product,
Albanese variety,
Albanese mapping,
symmetrized differential,
algebraic surface
Article copyright:
© Copyright 1982
American Mathematical Society