The Albanese mapping for a punctual Hilbert scheme. II. Symmetrized differentials and singularities
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- by Mark E. Huibregtse PDF
- Trans. Amer. Math. Soc. 274 (1982), 109-140 Request permission
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 \[ \varphi _n : H_X^n \stackrel {\sigma _n}{\to } X(n) \stackrel {f_n}{\to } A, \] 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 $C$.References
- Allen Altman and Steven Kleiman, Introduction to Grothendieck duality theory, Lecture Notes in Mathematics, Vol. 146, Springer-Verlag, Berlin-New York, 1970. MR 0274461
- Armand Borel, Linear algebraic groups, W. A. Benjamin, Inc., New York-Amsterdam, 1969. Notes taken by Hyman Bass. MR 0251042
- John Fogarty, Algebraic families on an algebraic surface, Amer. J. Math. 90 (1968), 511–521. MR 237496, DOI 10.2307/2373541 A. Grothendieck, Élements de géométrie algébrique, Inst. Hautes Études Sci. Publ. Math. 4 (1960); ibid 8 (1961); ibid 11 (1961); ibid 17 (1963); ibid 20 (1964); ibid 24 (1965); ibid 28 (1966); ibid 32 (1967). —, Fondaments de la ǵeométrie algébrique, (Extraits du Sém. Bourbaki 1957-1962), Sécretariat Mathématique, Paris, 1962.
- Mark E. Huibregtse, The Albanese mapping for a punctual Hilbert scheme. I. Irreducibility of the fibers, Trans. Amer. Math. Soc. 251 (1979), 267–285. MR 531979, DOI 10.1090/S0002-9947-1979-0531979-9
- Anthony A. Iarrobino, Punctual Hilbert schemes, Mem. Amer. Math. Soc. 10 (1977), no. 188, viii+112. MR 485867, DOI 10.1090/memo/0188
- Birger Iversen, Linear determinants with applications to the Picard scheme of a family of algebraic curves, Lecture Notes in Mathematics, Vol. 174, Springer-Verlag, Berlin-New York, 1970. MR 0292835
- Roy H. Ogawa, On the points of Weierstrass in dimensions greater than one, Trans. Amer. Math. Soc. 184 (1973), 401–417. MR 325997, DOI 10.1090/S0002-9947-1973-0325997-8
- A. Mattuck and A. Mayer, The Riemann-Roch theorem for algebraic curves, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 17 (1963), 223–237. MR 162798
- D. Mumford, Rational equivalence of $0$-cycles on surfaces, J. Math. Kyoto Univ. 9 (1968), 195–204. MR 249428, DOI 10.1215/kjm/1250523940 —, Abelian varieties, Oxford Univ. Press, Oxford, 1970.
- David Mumford, Lectures on curves on an algebraic surface, Annals of Mathematics Studies, No. 59, Princeton University Press, Princeton, N.J., 1966. With a section by G. M. Bergman. MR 0209285 A. A. Roitman, On $\Gamma$-equivalence of zero-dimensional cycles, Math. USSR-Sb. 15 (1971), 555-567.
- I. R. Shafarevich, Basic algebraic geometry, Die Grundlehren der mathematischen Wissenschaften, Band 213, Springer-Verlag, New York-Heidelberg, 1974. Translated from the Russian by K. A. Hirsch. MR 0366917
- Jean-Pierre Serre, Groupes algébriques et corps de classes, Publications de l’Institut de Mathématique de l’Université de Nancago, VII, Hermann, Paris, 1959 (French). MR 0103191 —, Morphismes universels et variété d’Albanese, Séminaire C. Chevalley, E. N. S. 1958/59.
- Kenji Ueno, Classification theory of algebraic varieties and compact complex spaces, Lecture Notes in Mathematics, Vol. 439, Springer-Verlag, Berlin-New York, 1975. Notes written in collaboration with P. Cherenack. MR 0506253
- Oscar Zariski, Introduction to the problem of minimal models in the theory of algebraic surfaces, Publications of the Mathematical Society of Japan, vol. 4, Mathematical Society of Japan, Tokyo, 1958. MR 0097403
Additional Information
- © Copyright 1982 American Mathematical Society
- 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