The tautological ring of the moduli space of curves. The moduli space of curves and its tautological ring ravi vakil the moduli space of curves has proven itself a central object in geometry. The recursion comes from the edge removal operation on the space of ribbon graphs. The case of the moduli space of curves with genus greater than two is tackled via the potential stability theorem. Since the stack am gx in this moduli problem on one hand is related to curves and on the other hand arises from the atwisted theory of a gauged linear sigma model, we will call it the atwisted moduli stack for curves from wittens glsm. Moduli of curves j harris, i morrison pdf alzaytoonah. Giesekers plan for using git to construct the moduli spaces of stable curves, now over 30 years old, has. The rough idea is but see the caveat below that the term moduli space is essentially a synonym for representing object and for classifying space. Giesekers plan for using git to construct the moduli spaces of stable curves, now over 30 years old, has recently been extended to moduli spaces of pointed stable curves and of stable maps by swinarski and baldwin.
Each family of stable curves induces a holomorphic maps into the moduli space. It turns out that the class of curves that can arise as limits of smooth curves are curves with only nodal singularities and nite automorphism group. People tend to say classifying space when in the context of topology, and they tend to say moduli space when in a context of complex geometry or algebraic geometry. Minicourse on moduli spaces university of michigan. For instance, the higherdimensional analogue of the moduli space of elliptic curves discussed above is the moduli space of abelian varieties, such as the siegel modular variety. Algebraic curves and their moduli spaces higher dimensional varieties and their moduli spaces gometric invariant theory and bridgeland stability minimal model program. Chaudhuri mpim operads and moduli of curves 7 august, 2014 5 36. We then consider the family of modular curves x 0p constructed by drinfeld, and we study their weierstrass points, a nite set of points of geometric interest.
Git constructions of moduli spaces of stable curves and maps. Since the stack am gx in this moduli problem on one hand is related to curves and on the other hand arises from the atwisted theory of a gauged linear sigma model, we will call it the atwisted moduli stack for curves from. The euler characteristic of the moduli space of curves. Moduli of curves and multiple roots emre can sertoz abstract wecompactifythemodulispaceofcurvesandmultiplesquareroots of a line bundle. Moduli of curves 2 wecontinuetousetheconventionsandtheabuseoflanguageintroducedinproperties of stacks, section 2. Feb 22, 2016 algebraic curves and their moduli spaces higher dimensional varieties and their moduli spaces gometric invariant theory and bridgeland stability minimal model program birational geometry and topology of mg moduli and degeneratons of algebraic curves via tropical geometry.
Don zagier, john harer, the euler characteristic of the moduli space of curves, inventiones mathematicae 1986 volume. The torelli group tg is the group of isotopy classes of di eomorphisms of a compact orientable surface. T ur curves by the universal property of the quotient ur, see groupoids in spaces, lemma 22. Unless otherwise mentioned our base scheme will be specz. For an html and pdf or ppt file with the same name, the html is an exposition. 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. Moduli of curves 3 conversely,letxtbeafamilyofcurves. Some notes describing the picard group of the moduli space of curves.
Mumford and deligne 1 introduced the concept of stable curve in their proof of the. Don zagier, john harer, the euler characteristic of the moduli space of. Cimatictp school on moduli of curves, february 29march 4, 2016 in guanajuato, mexico. For instance, the higherdimensional analogue of the moduli space of elliptic curves discussed. Lectures on the moduli stack of vector bundles on a curve. An introduction to moduli spaces of curves and its. Rational curves in the moduli of supersingular k3 surfaces. Knudsen established the existence of a projective moduli space for stable curves. Zagier department of mathematics, university of maryland, college park, md 20742, usa let fg 1, g 1, be the. The euler characteristic of the moduli space of curves j. Section 1 is an informal introduction to moduli spaces of smooth and stable curves. The moduli spaces of holomorphic maps elenynicoleta ionel abstract this is a draft of some informal notes that are meant to accompany the second week of the. Zagier department of mathematics, university of maryland, college park, md 20742, usa let fg 1, g 1, be the mapping class group consisting of all isotopy classes of basepoint and orientation preserving homeomorphisms of a closed, oriented. This is used to prove results that generalize the classical franchetta conjecture to the generic curve of genus g with n marked points and a levell structure.
Thestackofcurves 0dmj thissectionisthecontinuationofquot,section15. I added a few words outlining other work of harer on the stability of the cohomology of the moduli space of curves. The geometry of moduli spaces of pointed curves, the. The drawing signature pencil tool is very useful for the tablet pc since you can treat a pdf document just like a normal printed document and draw on the pc with the mouse or on the tablet pc using the pen. Please visit nakamura04 if you are interested in, since it could be still be a good introduction to the subject as. Pdf rational curves in the moduli of supersingular k3. We introduce the moduli space of genus, pointed curves, with enough information to give a feel. A boundary divisorial basis and its tree representation 27 8. Zagier the modular group takes its name from the fact that the points of the quotient space. Torelli groups and geometry of moduli spaces of curves richard m. Cimpacimatictp school on moduli of curves smr 2870 22. Other good introductions to moduli spaces include 10 and 20.
U,tr determinesa2arroww sw tsatisfyingthecocyclecondition. The euler characteristic of the moduli space of curves 467. The past decade has seen substantial progress in understanding the moduli space of curves, involvingideas,forexample,fromgeometryalgebraic,symplectic,anddifferential,physics, topology, and combinatorics. 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.
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. There is a strati cation of the moduli space using their topological type which is completely determined by the dual graph. 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. The aim of the course was to use this example to introduce the notion of algebraic stacks and to illustrate how. These are lecture notes of a short course on the moduli stack of vector bundles on an algebraic curve. You should also read chapter 5 and 6f of harris, j.
Letcurves bethestackwhose category of sections over a scheme sis the category of families of curves over s. Modular curves here derive from the semidirect product of z2 acting through multiplication by 1 on z. Pdf rational curves in the moduli of supersingular k3 surfaces. The authors show indeed in the next chapter that the moduli space can be. The goal of this article is to motivate and describe how gromovwitten theory can and has. Preface aims theaimofthisbookistoprovideaguidetoarichandfascinatingsubject. Introduction the forms of the relationships expressing shear modulus and damping ratio as a function of shear strain play an important role in determining the results of ground response analyses. By smooth curve, we mean a compact also known as proper or complete, smooth also known as nonsingular complex curve, i. Some notes about the kodaira dimension of the moduli space of curves. Dynamic moduli and damping ratios for cohesive soils by joseph i. Cimpacimatictp school on moduli of curves smr 2870. There is a strati cation of the moduli space using their topological type which is completely determined by the dual. The aim of this paper is to show how facts about the homology of the torelli group imply.
Faber in the period 19912011with continually improving. Drinfeld modular forms modulo p and weierstrass points on. You can just write or draw your signatures, ideas and thoughts onto the existing document. Summer school on moduli of curves and gromovwitten. Torelli groups and geometry of moduli spaces of curves. The resulting moduli stacks of smooth or stable genus g curves with n marked points are denoted, or. Fields of moduli of some special curves article pdf available in journal of pure and applied algebra 2201 may 2015 with 73 reads how we measure reads. In higher dimensions, moduli of algebraic varieties are more difficult to construct and study. Jul 30, 2015 rational curves in the moduli of supersingular k3 surfaces.
Then in the last chapter they show the reverse, that the properties of various moduli spaces can be proven using the techniques introduced in the book, such as the irreducibility of the moduli space, the diaz result that complete subvarieties of the moduli space have dimension at most genus 2, and moduli of hyperelliptic curves and severi. The tautological ring of the moduli space of curves m g is a subring rm g of the chow ring am g. Introduction the forms of the relationships expressing shear. All curves with at least knodes form a subvariety of codimension k. As moduli space towers they exhibit a frattini property, based on their monodromy groups as covers of the jline. The complex analytic structure and the relation to teichmuller space is further discussed in. Shingtung yau geometry of the moduli space of curves. Git constructions of moduli spaces of stable curves and maps ian morrison abstract. This subvariety is completely contained in the boundary.
As an application we obtain a new proof of the kontsevich constants for the ratio of the euclidean and the symplectic volumes of the moduli space of curves. A case of particular interest is the moduli stack m. The orbifold euler characteristic of the moduli space of curves was originally computed in. Using the goals of serres ladic representations book i will treat, in parallel, two cases of general ideas. We will focus on the moduli space of smooth curves and how much of the recent progress in understanding it has come through enumerative invariants in gromovwitten theory, something which we take date. The geometry of moduli spaces of pointed curves, the tensor. 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. For example, a curve in the moduli space might be a map a1. On atwisted moduli stack for curves from wittens gauged. These are lecture notes of a short course on the moduli stack of.
For example, a curve in the moduli space might be a map a 1. Lectures on the moduli stack of vector bundles on a curve jochen heinloth preliminary version, january 29, 2009 abstract. Stanford university, department of mathematics, building 380, stanford, california 94305, usa. It turns out that the class of curves that can arise as limits of smooth curves are curves with only nodal singularities and nite. There exists a positive constant c1, independent of g, such. The torelli group tg is the group of isotopy classes of di eomorphisms of a compact orientable surface of genus g that act trivially on the homology of the surface. A version of schwarz lemma that i proved, gave very. Hesse cubic curves the major part of this note is taken from nakamura04.
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