The Moduli Space of Curves

Dr. Carel Faber ist am Max-Planck-Institut für Mathematik in Bonn tätig.

This generalization of geometry is bound to have wide spread repercussions for mathematics as well as physics. The unearthing of it will entail a new golden age in the interaction of mathematics and physics. E. Witten (1986) The idea that the moduli space Mg of curves of fixed genus 9 - that is, the algebraic variety that parametrizes all curves of genus 9 - is an intriguing object in its own right seems to have come slowly. Although the para meters or moduli of curves surface in Riemann's famous memoir on abelian functions (from 1857) and in work of Hurwitz and later were considered by the geometers of the Italian school, for a long time they attracted attention only in the special case 9 = 1, where they were studied in the framework of the theory of modular functions. The work of Grothendieck, who in the early sixties pointed the way towards the right approach, and the subsequent construction (in 1965) of the moduli space Mg by Mumford were the first foundational work, to be followed by the construction of a compactification Mg by Deligne and Mumford in 1969. The theorem of Harris and Mumford saying that for 9 sufficiently large the space Mg is of general type was the first big insight in its structure.

The moduli space Mg of curves of fixed genus g - that is, the algebraic variety that parametrizes all curves of genus g - is one of the most intriguing objects of study in algebraic geometry these days. Its appeal results not only from its beautiful mathematical structure but also from recent developments in theoretical physics, in particular in conformal field theory.

Distribution of rational points and Kodaira dimension of fiber products.- How many rational points can a curve have?.- Quantum cohomology of rational surfaces.- Quantum intersection rings.- Mirror symmetry and elliptic curves.- A generalized Jacobi theta function and quasimodular forms.- Boundary behaviour of Hurwitz schemes.- Operads and moduli spaces of genus 0 Riemann surfaces.- Resolution of diagonals and moduli spaces.- The Chow ring of the moduli space of curves of genus 5.- The cohomology of algebras over moduli spaces.- Enumeration of rational curves via torus actions.- Cellular decompositions of compactified moduli spaces of pointed curves.- Generating functions in algebraic geometry and sums over trees.- Holomorphicity and non-holomorphicity in N = 2 supersymmetric field theories.- An arithmetic problem in surface geometry.- An orbifold partition of$$bar M_g^n$$.- Moduli of curves with non-abelian level structure.- Q-structure of conformal field theory with gauge symmetries.- On the cohomology of moduli spaces of rank two vector bundles over curves.