Carel Faber – författare

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.

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.

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.

This bookprovides an overview of the latest developments concerning the moduli of K3surfaces. It is aimed at algebraic geometers, but is also of interest to numbertheorists and theoretical physicists, and continues the tradition of relatedvolumes like “The Moduli Space of Curves” and “Moduli of Abelian Varieties,”which originated from conferences on the islands Texel and Schiermonnikoog andwhich have become classics.K3 surfacesand their moduli form a central topic in algebraic geometry and arithmeticgeometry, and have recently attracted a lot of attention from bothmathematicians and theoretical physicists. Advances in this field often resultfrom mixing sophisticated techniques from algebraic geometry, lattice theory,number theory, and dynamical systems. The topic has received significantimpetus due to recent breakthroughs on the Tate conjecture, the study ofstability conditions and derived categories, and links with mirror symmetry andstring theory. At the sametime, the theory of irreducible holomorphicsymplectic varieties, the higher dimensional analogues of K3 surfaces, hasbecome a mainstream topic in algebraic geometry.Contributors:S. Boissière, A. Cattaneo, I. Dolgachev, V. Gritsenko, B. Hassett, G. Heckman,K. Hulek, S. Katz, A. Klemm, S. Kondo, C. Liedtke, D. Matsushita, M.Nieper-Wisskirchen, G. Oberdieck, K. Oguiso, R. Pandharipande, S. Rieken, A. Sarti, I.Shimada, R. P. Thomas, Y. Tschinkel, A. Verra, C. Voisin.

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.