Intersection theory on the moduli space of curves and the matrix airy function. The arithmetic study of the moduli spaces began with jacobis fundamenta nova in 1829, and the modern theory was erected by. The moduli space of curves alessio corti october 27, 1997 this is a write up of my lecture in the cambridge \geometry seminar, an introduction to the construction and proof that the compacti ed moduli space m gof curves of genus gis a projective variety. You should also read chapter 5 and 6f of harris, j. Moduli of rank2 vector bundles, theta divisors, and the geometry of curves in projective space. Minicourse on moduli spaces university of michigan. Perhaps, it is so, because very little is known about it, while on the other hand, so much is known about algebraic curves, which form the points of the moduli spaces. Pdf a tourists guide to intersection theory on moduli spaces of. Don zagier, john harer, the euler characteristic of the moduli space of curves, inventiones mathematicae 1986 volume. The orbifold euler characteristic of the moduli space of curves was originally computed in. We prove that the moduli spaces of rational curves of degree at most 3 in linear sections of the grassmannian g r 2, 5 are all.
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. Moduli of curves j harris, i morrison pdf alzaytoonah. Continuity of the map from the universal family into the moduli space garantees that curves which are almost similar will correspond to points in the moduli space which are close. On the picard group of the moduli scheme of stable curves. Some notes about the kodaira dimension of the moduli space of curves. Chapter 1 is that if the mth hilbert point is semistable, then the curve is semistable as a curve. An introduction to moduli spaces of curves and its. To apply for support to attend the conference, contact.
Introduction one of the most fundamental objects of study in algebraic geometry is the study of the moduli curves. The complex analytic structure and the relation to teichmuller space is further discussed in. We compute the number of moduli of all irreducible components of the moduli space of smooth curves on enriques surfaces. Just as it sounds, bootyful beginnings is an excellent workout program for athletes that many not have much familiarity with the movements needed for the workout program. We treat the problem of completing the moduli space for roots of line bundles on curves.
I added a few words outlining other work of harer on the stability of the cohomology of the moduli space of curves. The most recent draft of these notes may be downloaded at modulimoristablecurves. Their goal is to introduce and motivate basic concepts and constructions such as orbifolds and stacks important in the study of moduli spaces of curves and abelian varieties through the example of elliptic curves. Their goal is to introduce and motivate basic concepts and constructions such as orbifolds and stacks important in the study of moduli spaces of curves and abelian varieties through the example of.
In higher dimensions, moduli of algebraic varieties are more difficult to construct and study. Geometry of algebraic curves volume ii with a contribution. Knudsen established the existence of a projective moduli space for stable curves. The past decade has seen substantial progress in understanding the moduli space of curves, involvingideas,forexample,fromgeometryalgebraic,symplectic,anddifferential,physics, topology, and combinatorics. Proceedings of the international symposium on algebraic geometry, kyoto, 1977 15. Q, and there is a bijection between such maps and families over a1. Preface aims theaimofthisbookistoprovideaguidetoarichandfascinatingsubject. When g 1, this stack may be compactified by adding new boundary points which correspond to stable nodal curves together with their isomorphisms. The second rational homology group of the moduli space of. Originally, these varieties were introduced by severi in the plane case in order to prove the irreducibility of the moduli spaces of curves mg in characteristic zero. The moduli space is the classifying space of all curves, and the existence of a universal family makes sure that the classification remembersthe structure of the curves.
A moduli stack of elliptic curves is a moduli stack of elliptic curves, hence a stack. Moduli of curves 3 conversely,letxtbeafamilyofcurves. Some notes describing the picard group of the moduli space of curves. Using the method of algebraic stacks, we can prove not only the irreducibility of mg itself, but of all higher level moduli spaces of curves too cf. Sarah brodsky, michael joswig, ralph morrison and bernd sturmfels. Looijenga and others published a minicourse on moduli of curves find, read and cite all the research you need on researchgate. Finally, we describe some of what is known about the tautological ring. The slope stability of the logarithmic cotangent bundle of the dm moduli spaces, chern number inequality and other properties will follow. The arithmetic study of the moduli spaces began with jacobis fundamenta nova in 1829, and the modern theory was erected by eichlershimura, igusa, and delignerapoport. S desigularization of moduli varieties of vector bundles on curves. These informal notes are an expanded version of lectures on the moduli space of elliptic curves given at zhejiang university in july, 2008.
Author links open overlay panel ciro ciliberto a thomas dedieu b concettina galati c. Shingtung yau geometry of the moduli space of curves. We construct the generic component of the moduli space of the germs of legendrian curves with generic plane projection topologicaly equivalent to a curve y n x m. An introduction to moduli spaces of curves and its intersection theory 5 example 1. Each family of stable curves induces a holomorphic maps into the moduli space. We next introduce the tautological cohomology or chow classes.
For example, a curve in the moduli space might be a map a1. Moduli spaces of algebraic curves of genus zero the moduli spaces of algebraic curves have been a very exciting object to study in algebraic geometry. To solve the moduli problem for curves we need to find a geometric object, denoted the universal family, a smaller geometric object, called the moduli space, and a good mapping from the universal family into the moduli space. Mumford and deligne 1 introduced the concept of stable curve in their proof of the irreducibility of the moduli space of curves of genus g. Mori theory of moduli spaces of stable curves projective press. In particular we may consider the numerical equivalence for algebraic cycles of m p. On severi varieties and moduli spaces of curves in. Notes on the construction of the moduli space of curves dan edidin the purpose of these notes is to discuss the problem of moduli for curves of genus g. The modern technology is based on rst constructing the stack m g dm. Zariski 7 initiated the construction of the moduli of plane curve singu. The moduli space is constructed such that every point in the space corresponds to a certain class of curves, and vice versa, every class of curves is. Moduli of vector bundles on curves with parabolic structures. We introduce the moduli space of genus, pointed curves, with enough information to give a feel for its basic geography.
T ur curves by the universal property of the quotient ur, see groupoids in spaces, lemma 22. This is the problem underlying siegel modular form theory. Geometry of moduli spaces of rational curves in linear sections of. Special attention is devoted to higher spin curves within the. N is a ne moduli space for the moduli problem of nite sets up to bijection. S on the cohomology groups of moduli spaces of vector bundles on curves. The irreducibility of the space of curves of given genus. Stanford university, department of mathematics, building 380, stanford, california 94305, usa. The guiding philosophy, popularized by mumford, is that, not only it is fair game to ask about moduli spaces any question that is of interest for a general variety, but that. Pdf in the past few decades, moduli spaces of curves have become increasingly prominent and important in. Depending on the restrictions applied to the classes of algebraic curves considered, the corresponding moduli problem and the moduli space is different. Its authors are research mathematicians who have actively participated in the development of the geometry of algebraic curves. Depending on the restrictions applied to the classes of algebraic curves considered, the corresponding moduli problem and the moduli space is.
A version of schwarz lemma that i proved, gave very sharp geometric height inequalities in algebraic geometry. This work is a comprehensive treatment of recent developments in the study of elliptic curves and their moduli spaces. The perturbed ricci metric that we introduced has bounded. The aim is to provide a brief introduction to algebraic stacks, and then to give several. In chapter 2, we use the results of chapter 1 to give an indirect proof that the ncanonical embedding of a stable curve is stable if n. Indeed m p is the moduli scheme associated to the smooth moduli stack of stable genus 9 curves. The aim is to provide a brief introduction to algebraic stacks, and then.
1071 1559 443 1284 55 240 99 1481 1130 1621 392 1465 67 490 934 1011 371 81 886 611 522 1362 204 564 83 231 718 547 1073 248 136 1125 469 466 860 1097