Moduli of algebraic curves

In algebraic geometry, a moduli space of (algebraic) curves is a geometric space (typically a scheme or an algebraic stack) whose points represent isomorphism classes of algebraic curves. It is thus a special case of a moduli space. Depending on the restrictions applied to the classes of algebraic curves considered, the corresponding moduli problem and the moduli space is different. One also distinguishes between fine and coarse moduli spaces for the same moduli problem.

The most basic problem is that of moduli of smooth complete curves of a fixed genus. Over the field of complex numbers these correspond precisely to compact Riemann surfaces of the given genus, for which Bernhard Riemann proved the first results about moduli spaces, in particular their dimensions ("number of parameters on which the complex structure depends").

Moduli stacks of stable curves

The moduli stack ${\mathcal {M}}_{g}$ classifies families of smooth projective curves, together with their isomorphisms. When g > 1, this stack may be compactified by adding new "boundary" points which correspond to stable nodal curves (together with their isomorphisms). A curve is stable if it is complete, connected, has no singularities other than double points, and has only a finite group of automorphisms. The resulting stack is denoted ${\overline {\mathcal {M}}}_{g}$ . Both moduli stacks carry universal families of curves.

Both stacks above have dimension $3g-3$ ; hence a stable nodal curve can be completely specified by choosing the values of 3g-3 parameters, when g > 1. In lower genus, one must account for the presence of smooth families of automorphisms, by subtracting their number. There is exactly one complex curve of genus zero, the Riemann sphere, and its group of isomorphisms is PGL(2). Hence the dimension of ${\mathcal {M}}_{0}$ is

dim(space of genus zero curves) - dim(group of automorphisms) = 0 - dim(PGL(2)) = -3.

Likewise, in genus 1, there is a one-dimensional space of curves, but every such curve has a one-dimensional group of automorphisms. Hence, the stack ${\mathcal {M}}_{1}$ has dimension 0.

Coarse moduli spaces

One can also consider the coarse moduli spaces representing isomorphism classes of smooth or stable curves. These coarse moduli spaces were actually studied before the notion of moduli stack was invented. In fact, the idea of a moduli stack was invented by Deligne and Mumford in an attempt to prove the projectivity of the coarse moduli spaces. In recent years, it has become apparent that the stack of curves is actually the more fundamental object.

The coarse moduli spaces have the same dimension as the stacks when g > 1; however, in genus zero the coarse moduli space has dimension zero, and in genus one, it has dimension one.

Moduli of marked curves

One can also enrich the problem by considering the moduli stack of genus g nodal curves with n marked points, pairwise distinct and distinct from the nodes. Such marked curves are said to be stable if the subgroup of curve automorphisms which fix the marked points is finite. The resulting moduli stacks of smooth (or stable) genus g curves with n marked points are denoted ${\mathcal {M}}_{g,n}$ (or ${\overline {\mathcal {M}}}_{g,n}$ ), and have dimension 3g-3 + n.

A case of particular interest is the moduli stack ${\overline {\mathcal {M}}}_{1,1}$ of genus 1 curves with one marked point. This is the stack of elliptic curves. Level 1 modular forms are sections of line bundles on this stack, and level N modular forms are sections of line bundles on the stack of elliptic curves with level N structure (roughly a marking of the points of order N).

Boundary geometry

An important property of the compactified moduli spaces ${\overline {\mathcal {M}}}_{g,n}$ is that their boundary can be described in terms of moduli spaces ${\overline {\mathcal {M}}}_{g,n}$ for smaller g: Given a marked, stable, nodal curve one can associate its dual graph, a graph with vertices labelled by nonnegative integers and allowed to have loops, multiple edges and even numbered half-edges. Here the vertices of the graph correspond to irreducible components of the nodal curve, the labelling of a vertex is the arithmetic genus of the corresponding component, edges correspond to nodes of the curve and the half-edges correspond to the markings. The closure of the locus of curves with a given dual graph in ${\overline {\mathcal {M}}}_{g,n}$ is isomorphic to the stack quotient of a product $\prod _{v}{\overline {\mathcal {M}}}_{g_{v},n_{v}}$ of compactified moduli spaces of curves by a finite group. In the product the factor corresponding to a vertex v has genus g_v taken from the labelling and number of markings n_v equal to the number of outgoing edges and half-edges at v.

References

Grothendieck, Alexander (1960–1961). "Techniques de construction en géométrie analytique. I. Description axiomatique de l'espace de Teichmüller et de ses variantes" (PDF). Séminaire Henri Cartan 13 no. 1, Exposés No. 7 and 8. Paris: Secrétariat Mathématique.

Mumford, D.; Fogarty, J.; Kirwan, F. Geometric invariant theory. Third edition. Ergebnisse der Mathematik und ihrer Grenzgebiete (2) (Results in Mathematics and Related Areas (2)), 34. Springer-Verlag, Berlin, 1994. xiv+292 pp. MR1304906 ISBN 3-540-56963-4
Deligne, Pierre; Mumford, David (1969). "The irreducibility of the space of curves of given genus" (PDF). Publications Mathématiques de l'IHÉS. 36. Paris: 75–109. doi:10.1007/bf02684599.

Harris, Joe; Morrison, Ian (1998). Moduli of Curves. Springer Verlag. ISBN 0-387-98429-1.

Katz, Nicholas M; Mazur, Barry (1985). Arithmetic Moduli of Elliptic Curves. Princeton University Press. ISBN 0-691-08352-5.
Geometry of Algebraic Curves, Volume II, Arbarello Enrico, Cornalba Maurizio, Griffiths Phillip with a contribution by Joseph Daniel Harris. Series: Grundlehren der mathematischen Wissenschaften, Vol. 268, 2011, XXX, 963p. 112 illus., 30 illus. in color.

External links

The Moduli Space of Curves Resource Page