×

zbMATH — the first resource for mathematics

The geometry of the compactification of the Hurwitz scheme. (English) Zbl 0866.14013
The purpose of this paper is to study the geometry of the Harris-Mumford compactification of the Hurwitz scheme. The Hurwitz scheme parametrizes certain ramified coverings \(f:C \to\mathbb{P}^1\) of the projective line by smooth curves. Thus, from the very outset, one sees that there are essentially two ways to approach the Hurwitz scheme:
(1) We start with \(\mathbb{P}^1\) and regard the objects of interest as coverings of \(\mathbb{P}^1\).
(2) We start with \(C\) and regard the objects of interest as morphisms from \(C\) to \(\mathbb{P}^1\).
One finds that one can obtain the most information about the Hurwitz scheme and its compactification by exploiting interchangeably these two points of view. Our first main result is the following theorem.
Let \(b,d\), and \(g\) be integers such that \(b=2d+ 2g-2\), \(g\geq 5\) and \(d>2g+4\). Let \({\mathcal H}\) be the Hurwitz scheme over \(\mathbb{Z} [{1\over b!}]\) parametrizing coverings of the projective line of degree \(d\) with \(b\) points of ramification. Then \(\text{Pic} ({\mathcal H})\) is finite.
The number \(g\) is the genus of the “curve \(C\) upstairs” of the coverings in question. Note, however, that the Hurwitz scheme \({\mathcal H}\), and hence also the genus \(g\), are completely determined by \(b\) and \(d\). – Note that although in the statement of the theorem we spoke of “the” Hurwitz “scheme,” there are in fact several different Hurwitz schemes used in the literature, some of which are, in fact, not schemes, but stacks. The main idea of the proof is that by combinatorially analyzing the boundary of the compactification of the Hurwitz scheme, one realizes that there are essentially three kinds of divisors in the boundary, which we call excess divisors, which are “more important” than the other divisors in the boundary in the sense that the other divisors map to sets of codimension \(\geq 2\) under various natural morphisms. On the other hand, we can also consider the moduli stack \({\mathcal G}\) of pairs consisting of a smooth curve of genus \(g\), together with a linear system of degree \(d\) and dimension 1. The subset of \({\mathcal G}\) consisting of those pairs that arise from Hurwitz coverings is open in \({\mathcal G}\), and its complement consists of three divisors, which correspond precisely to the excess divisors. Using results of Harer on the Picard group of \({\mathcal M}_g\), we show that these three divisors on \({\mathcal G}\) form a basis of \(\text{Pic} ({\mathcal G}) \otimes_\mathbb{Z} \mathbb{Q}\), and in fact, we even compute explicitly the matrix relating these three divisors on \({\mathcal G}\) to a certain standard basis of \(\text{Pic} ({\mathcal G}) \otimes_\mathbb{Z} \mathbb{Q}\). The above theorem then follows formally.
Crucial to our study of the Hurwitz scheme is its compactification by means of admissible coverings and we prove a rather general theorem concerning the existence of a canonical logarithmic algebraic stack \(({\mathcal A}, M)\) parametrizing such coverings:
Fix non-negative integers \(g,r,q,s,d\) such that \(2g-2+r =d(2q-2+s) \geq 1\). Let \({\mathcal A}\) be the stack over \(\mathbb{Z}\) defined as follows: For a scheme \(S\), the objects of \({\mathcal A}(S)\) are admissible coverings \(\pi:C\to D\) of degree \(d\) from a symmetrically \(r\)-pointed stable curve \((f:C\to S\); \(\mu_f \subseteq C)\) of genus \(g\) to a symmetrically \(s\)-pointed stable curve \((h:D \to S\); \(\mu_h \subseteq D)\) of genus \(q\); and the morphisms of \({\mathcal A} (S)\) are pairs of \(S\)-isomorphisms \(\alpha: C\to C\) and \(\beta: D\to D\) that stabilize the divisors of marked points such that \(\pi\circ \alpha= \beta\circ \pi\). Then \({\mathcal A}\) is a separated algebraic stack of finite type over \(\mathbb{Z}\). Moreover, \({\mathcal A}\) is equipped with a canonical log structure \(M_{\mathcal A} \to {\mathcal O}_{\mathcal A}\), together with a logarithmic morphism \(({\mathcal A}, M_{\mathcal A}) \to \overline {{\mathcal M} {\mathcal S}}^{\log}_{q,s}\) (obtained by mapping \((C;D;\pi) \mapsto D)\) which is log étale (always) and proper over \(\mathbb{Z} [{1\over d!}]\).

MSC:
14H10 Families, moduli of curves (algebraic)
14C22 Picard groups
14E20 Coverings in algebraic geometry
PDF BibTeX Cite
Full Text: DOI
References:
[1] Arbarello, E. and Cornalba, M., The Picard Groups of the Moduli Space of Curves, Topology, 2 6 (1987), 153-171. · Zbl 0625.14014
[2] Arbarello, E., Weierstrass Points and the Moduli of Curves, Compositio Math., 29 (1974), 325-342. · Zbl 0355.14013
[3] Artin, M., Algebrization of Formal Moduli, I, Global Analysis (Papers in Honor of K. Kodaira), University of Tokyo Press, 1969, 21-71. · Zbl 0205.50402
[4] Clebsch, A., Ziir Theorie der Riemann’schen Flache, Math. Ann., 6 (1872), 216-230.
[5] Grothendieck, A., Geometric Formelle et Geometric Algebrique, Seminaire Bourbaki 182.
[6] Diaz, S. and Donagi, R., Hurwitz Surfaces with Nontrivial Divisors, Algebraic Geometry, Sundance 1988, Contemp. Math., Amer. Math. Soc., 116 (1991), 1-8. · Zbl 0752.14005
[7] Diaz, S., Donagi, R. and Harbater, D., Every Curve is a Hurwitz Space, Duke J. Math., 59 (1989), 737-746. · Zbl 0712.14013
[8] Deligne, P. and Mumford, D., The Irreducibility of the Space of Curves of Given Genus, Institute des Hautes Etudes Scientifiques Publications Mathematiques, 36 (1969), 75-109. · Zbl 0181.48803
[9] Grothendieck, A. and Dieudonne, J., Etude Locale des schemas et des morphismes de schemas, Publ. Math. IHES, 20 (1964), 24 (1965), 28 (1966), 32 (1967).
[10] Fallings, G., Endlichkeitssatze fur Abelschen Varietaten liber Zahlkorpern, Inv. Math., 73 (1983), 349-366.
[11] Fallings, G. and Chai, C. L., Degenerations of Abelian Varieties, Springer-Verlag, 1990, Chapter 1.
[12] Grothendieck, A., Fondements de la Geometric Algebrique, Seminaire Bourbaki 1957-62, Secretariat Math., Paris, 1962.
[13] Fulton, W., Hurwitz Schemes and the Irreducibility of Moduli of Algebraic Curves, Annals of Math., 90 (1969), 542-575. · Zbl 0194.21901
[14] Griffiths, P. and Harris, L, Algebraic Geometry, Wiley-lntej:sciencQ, 1978, pp. 193-211.
[15] Girard, J., Cohomologie non abelienne, Springer-Verlag. 1971.
[16] Harer, J., The Second Homology Group of the Mapping Class Group of an Orientable Surface, Inv. Math., 72 (1982), 221-239. · Zbl 0533.57003
[17] , The Cohomology of the Moduli Space of Curves, Theory of Moduli (Montecatini Terme, 1985), E. Sernesi (Ed.), Lecture Notes in Math., Springer-Verlag, 1337 (1988).
[18] Harris, J. and Diaz, S., The Geometry of the Severi Variety II: Independence of Divisor Classes and Examples, Algebraic Geometry (Sundance, UT, 1986), Lecture Notes in Math., Springer- Verlag, 1311 (1988), 23-50. · Zbl 0677.14004
[19] Harris, J. and Mumford, D., On the Kodaira Dimension of the Moduli Space of Curves, Inv. Math. ,67 (1982), 23-86. · Zbl 0506.14016
[20] Hurwitz, A., Uber Riemann’schen Flache mit gegebenen Verzweigungs-punkten, Math. Ann., 39 (1891), 1-61.
[21] Kato, K., Logarithmic Structures of Fontaine-Illusie, Proceedings of the First JAMI Conference, Johns-Hopkins University Press, 1990, 191-224. · Zbl 0776.14004
[22] Knudsen, F. F., The Projectivity of the Moduli Space of Stable Curves, II: The Stacks M^, Math. Scand., 52 (1983), 161-199. · Zbl 0544.14020
[23] , The Projectivity of the Moduli Space of Stable Curves, III: The Line Bundles on M^ n, and a Proof of the Projectivity of M^ n in Characteristic 0, Math. Scand., 52 (1983), 200-212. · Zbl 0544.14021
[24] Lang, S., Introduction to Arakelov Theory, Springer-Verlag, 1988, Appendix by Vojta. · Zbl 0667.14001
[25] Liiroth, J., Uber Verzweigungsschnitte und Querschnitte in einer Riemannschen Flache, Math. A n n . , 4 (1871), 181-184.
[26] Moret-Bailly, L., Expose II, in Seminaire sur Les Pinceaux Arithmetiques: La Conjecture de Mordell, edited by L. Szpiro, Asterisque, 127 (1985).
[27] Fulton, W., On the Irreducibility of the Moduli Space of Curves, Inv. Math., 67 (1982), 87-88. · Zbl 0506.14017
[28] Milne, J. S., Etale Cohomology, Princeton University Press, 1980.
[29] Grothendieck, A. et al., Revetements etales et Groupe Fondamental, Lecture Notes in Math., Springer-Verlag, 224 (1971).
[30] Vojta, P., Diophantine Approximations and Value Distribution Theory, Lecture Notes in Math., Springer-Verlag, 1239 (1987). · Zbl 0609.14011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.