Towards an Enumerative Geometry of the Moduli Space of Curves

@inproceedings{Mumford1983TowardsAE,
  title={Towards an Enumerative Geometry of the Moduli Space of Curves},
  author={David Mumford},
  year={1983}
}
The goal of this paper is to formulate and to begin an exploration of the enumerative geometry of the set of all curves of arbitrary genus g. By this we mean setting up a Chow ring for the moduli space M g of curves of genus g and its compactification M g, defining what seem to be the most important classes in this ring and calculating the class of some geometrically important loci in M g in terns of these classes. We take as a model for this the enumerative geometry of the Grassmannians. Here… Expand
A CALCULUS FOR THE MODULI SPACE OF CURVES
The goal of this paper is to formulate and to begin an exploration of the enumerative geometry of the set of all curves of arbitrary genus g. By this we mean setting up a Chow ring for the moduliExpand
Geometric cycles on moduli spaces of curves
The aim of this thesis is the explicit computation of certain geometric cycles in moduli spaces of curves. In recent years, divisors of Mg,n have been extensively studied. Computing classes inExpand
Divisors in the moduli spaces of curves
The calculation by Harer [12] of the second homology groups of the moduli spaces of smooth curves over C can be regarded as a major step towards the understanding of the enumerative geometry of theExpand
The cohomology of the moduli space of curves
The purpose of these notes is to give an exposition of recent work of several people on the topology and geometry of the moduli space of curves. Moduli space may be approached in many different ways.Expand
On the tautological ring of M g , n
In this section, we briefly describe the objects under consideration for the sake of non-experts. A more detailed informal exposition of these well-known ideas is given in [PV]. When studying RiemannExpand
Intersection-theoretical Computations on M G
Introduction. In this paper we explore several concrete problems, all more or less related to the intersection theory of the moduli space of (stable) curves, introduced by Mumford [Mu 1]. In §1 weExpand
ON THE SLOPES OF THE MODULI SPACES OF CURVES
Let Mg be the moduli space of smooth curves of genus g and Mg the moduli space of stable curves of genus g. Then Mg = MgU A, where A = YJg£ ] A*> Ao is the closure of the locus of genus g — 1 curvesExpand
On the tautological ring of Mg,n
In this section, we briefly describe the objects under consideration for the sake of non-experts. A more detailed informal exposition of these well-known ideas is given in [PV]. When studying RiemannExpand
Geometric Proofs of Some Results of Morita Geometric Proofs of Some Results of Morita
which give relations between certain two dimensional cohomology classes of various moduli spaces of curves. We have reformulated Morita's results in more geometric language to facilitate theirExpand
Towards an intersection theory on Hurwitz spaces
Moduli spaces of algebraic curves are closely related to Hurwitz spaces, that is, spaces of meromorphic functions on curves. All of these spaces naturally arise in numerous problems of algebraicExpand
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 25 REFERENCES
The second homology group of the mapping class group of an orientable surface
In I-7] Mumford shows that the Picard group P ic (~ ' ) is isomorphic to H2(F; 2~) and conjectures the latter is rank one, g>3 . We prove this below for g>5 . Another interpretation of this theoremExpand
Toward a numerical theory of ampleness
Introduction Chapter I. Intersection Numbers ? 1. The polynomial theorem of Snapper ? 2. The definition and some properties of intersection numbers ? 3. Degrees and Hilbert polynomials ? 4.Expand
The Yang-Mills equations over Riemann surfaces
  • M. Atiyah, R. Bott
  • Mathematics
  • Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences
  • 1983
The Yang-Mills functional over a Riemann surface is studied from the point of view of Morse theory. The main result is that this is a ‘perfect' functional provided due account is taken of its gaugeExpand
Riemann-roch for singular varieties
The basic tool for a general Riemann-Roch theorem is MacPherson’s graph construction, applied to a complex E. of vector bundles on a scheme Y, exact off a closed subset X. This produces a localizedExpand
FAMILIES OF ALGEBRAIC CURVES WITH FIXED DEGENERACIES
In this paper we prove that there exist only finitely many nonisomorphic and nonconstant curves of fixed genus, defined over a fixed function field and having bad reductions at a given finite set ofExpand
A new compactification of the Siegel space and degeneration of Abelian varieties. II
This is a continuation of the article with the same title appeared in this journal, to the introduction of which we referr the reader for the summary of contents of this article.
The irreducibility of the space of curves of given genus
© Publications mathematiques de l’I.H.E.S., 1969, tous droits reserves. L’acces aux archives de la revue « Publications mathematiques de l’I.H.E.S. » (http://www.Expand
Weierstrass points and moduli of curves
© Foundation Compositio Mathematica, 1974, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditionsExpand
Rational equivalence on singular varieties
© Publications mathématiques de l’I.H.É.S., 1975, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http://Expand
Projectivity of the moduli space of stable curves
A self-contained mechanism for the preparation handling and application of liquified materials such as coating compounds, joint sealers, crack fillers, waterproofing compounds and the like, isExpand
...
1
2
3
...