Moduli Of Curves With Principal And Spin Bundles
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Moduli of Curves with Principal and Spin Bundles
The moduli space Mgbar of genus g stable curves is a central object in algebraic geometry. From the point of view of birational geometry, it is natural to ask if Mgbar is of general type. Harris-Mumford and Eisenbud-Harris found that Mgbar is of general type for genus g>=24 and g=22. The case g=23 keep being mysterious. In the last decade, in an attempt to clarify this, a new approach emerged: the idea is to consider finite covers of Mgbar that are moduli spaces of stable curves equipped with additional structure as l-covers (l-th roots of the trivial bundle) or l-spin bundles (l-th roots of the canonical bundle). These spaces have the property that the transition to general type happens to a lower genus. In this work we intend to generalize this approach in two ways: - a study of moduli space of curves with any root of any power of the canonical bundle; - a study of the moduli space of curves with G-covers for any finite group G. In order to define these moduli spaces we use the notion of twisted curve (see Abramovich-Corti-Vistoli). The fundamental result obtained is that it is possible to describe the singular locus of these moduli spaces via the notion of dual graph of a curve. Thanks to this analysis, we are able to develop calculations on the tautological rings of the spaces, and in particular we conjecture that the moduli space of curves with S3-covers is of general type for odd genus g>=13.
Compact Moduli Spaces and Vector Bundles
Author: Valery Alexeev
language: en
Publisher: American Mathematical Soc.
Release Date: 2012
This book contains the proceedings of the conference on Compact Moduli and Vector Bundles, held from October 21-24, 2010, at the University of Georgia. This book is a mix of survey papers and original research articles on two related subjects: Compact Moduli spaces of algebraic varieties, including of higher-dimensional stable varieties and pairs, and Vector Bundles on such compact moduli spaces, including the conformal block bundles. These bundles originated in the 1970s in physics; the celebrated Verlinde formula computes their ranks. Among the surveys are those that examine compact moduli spaces of surfaces of general type and others that concern the GIT constructions of log canonical models of moduli of stable curves. The original research articles include, among others, papers on a formula for the Chern classes of conformal classes of conformal block bundles on the moduli spaces of stable curves, on Looijenga's conjectures, on algebraic and tropical Brill-Noether theory, on Green's conjecture, on rigid curves on moduli of curves, and on Steiner surfaces.
Moduli of Curves and Abelian Varieties
Author: Carel Faber
language: en
Publisher: Springer Science & Business Media
Release Date: 2012-12-06
The present volume, with contributions of R. Dijkgraaf, C. Faber, G. van der Geer, R. Rain, E. Looijenga, and F. Oort, originates from the Dutch Intercity Seminar on Moduli (year 1995-96). Some of the articles here were discussed, in preliminary form, in the seminar; others are completely new. Two introductory papers, on moduli of abelian varieties and on moduli of curves, accompany the articles. Topics include a stratification of a moduli space of abelian varieties in positive characteristic, and the calculation of the classes of the strata, tautological classes for moduli of abelian varieties as well as for moduli of curves, correspondences between moduli spaces of curves, locally symmetric families of curves and jaco bians, and the role of symmetric product spaces in quantum field theory, string theory and matrix theory. This Intercity Seminar is part of the long term project "Algebraic curves and Riemann surfaces: geometry, arithmetic and applications" , sponsored hy the Netherlands Organization for Scientific Research (NWO), that has been running since 1994. Its ancestry can be traced back to joint activities in the seventies (if not earlier), which as of 1980 had evolved into active biweekly research seminars. These have been a focal point of Dutch algebraic geometry and singularity theory since. We are grateful to NWO for its support for the project. C.F. thanks the Max-Planck-Institut fur Mathematik, Bonn, for support during the final stages of the preparation of this volume.