What constitutes reasonable is hard to say in general, but it can include such disparate examples as 1. The chenruan cohomology of moduli of curves of genus 2. Geometric invariant theory and moduli spaces of pointed curves. Nonsimply laced cases, international mathematics research notices, vol. Mumford, in towards an enumerative geometry of the moduli space of curves publishedin1983,writes. Modular curves september 4, 20 the rst examples of shimura varieties we encounter are the modular curves. Other good introductions to moduli spaces include 18 and 33. Elliptic curves over schemes the notion of elliptic curves over arbitrary schemes is indispensable for the topic of moduli spaces. The weilpetersson volume grows exponentially in g see also 23, in particular, a rational multiple of. This thesis consists of a study of elliptic curves, and of elliptic surfaces. Since nefness is a condition that only depends on curves in the space, concretely, this condition can be checked in moduli situations by considering families over a 1dimensional base. One can prove that e\o can be mapped isomorphically to a curve in.
In this seminar, we will read selected articles of the handbook of moduli. 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. The moduli space m g 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. Moduli of curves and multiple roots emre can sertoz abstract wecompactifythemodulispaceofcurvesandmultiplesquareroots of a line bundle. There will be a preliminary discussion on thursday, 22. Moduli of curves 3 conversely,letxtbeafamilyofcurves. This note is about invariants of moduli spaces of curves. Git constructions of moduli spaces of stable curves and maps.
Combinatorial and algebrogeometric cohomology classes on. You can just write or draw your signatures, ideas and thoughts onto the existing document. Finally, we are not going to talk about curves over elds that are not algebraically closed. In this paper, we investigate the chenruan cohomology ring of the moduli space of smooth genus 2 curves with marked points, and of the moduli space of stable genus 2 curves with marked points. Moduli of families of curves for conformal and quasiconformal mappings. There has been a huge amount of work on algebraic curves.
Section1is an informal introduction to moduli spaces of smooth and stable curves. Chapter 1 is that if the mth hilbert point is semistable, then the curve is semistable as a curve. Ian morrison and michael thaddeus abstract the main result of this dissertation is that hilbert points parametrizing smooth curves with marked points are gitstable with respect to a wide range of linearizations. Supersymmetric curves are the analogue of riemann surfaces in super geometry. Coherent sheaves on primitive multiple curves and their moduli. Integrable systems and moduli spaces of curves 5 1. The resulting master curve of the modulus, as a function of time, formed in this manner. The moduli space of curves and its tautological ring ravi vakil t he moduli space of curves has provenitself a central object in geometry. Moduli of families of curves and extremal partitions. In higher dimensions, moduli of algebraic varieties are more difficult to construct and study. We establish some foundational results about complex delignemumford superstacks, and we then prove that the moduli superstack of supersymmetric curves is a smooth complex delignemumford superstack. I have multiple curves that i want to combine into 1 object, an icon to represent my chararcters foot.
The orbifold euler characteristic of the moduli space of curves was originally computed in. Pdf rational curves in the moduli of supersingular k3 surfaces. We study in particular the onedimensional substack of curves with at least 3g4. Our curves will be connected unless we especially describe them as possibly disconnected. 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.
The perturbed ricci metric that we introduced has bounded. Thus it is natural to ask how the curvatures vary in terms of g. Thestackofcurves 0dmj thissectionisthecontinuationofquot,section15. An introduction to moduli spaces of curves 2 parameters. 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 resulting moduli space is smooth and the parametrized objects have a goodmoduliinterpretation. The slope stability of the logarithmic cotangent bundle of the dm moduli spaces, chern number inequality and other properties will follow. Unless otherwise mentioned our base scheme will be specz. U,tr determinesa2arroww sw tsatisfyingthecocyclecondition. 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. M0,n be a stable nmarked curve of genus 0 and let q1. Given that curves in projective space can become arbitrarily singular, it is an amazing fact that the moduli space of curves can be compacti. We introduce the moduli space of genus, pointed curves, with enough information. We will introduce the minimal notions that will be used in what follows and assume a certain familiarity. Combinatorial and algebrogeometric cohomology classes on the moduli spaces of curves enrico arbarello 1and maurizio cornalba,2 dedicated to the memory of claude itzykson. The complex analytic structure and the relation to teichmuller space is further discussed in.
Moduli of curves anand deopurkar consider a reasonable class of objects in algebraic geometry. Torelli groups and geometry of moduli spaces of curves. The arithmetic study of the moduli spaces began with jacobis fundamenta nova in 1829, and the modern theory was erected. Algebraic geometery the moduli space of tropical curves can be used to explicitly describe the. Its cohomology ring has a elegant structure, and mumford suggested studying the moduli space of curves in the same way. The moduli space m g of complete nonsingular curves of genus g admits a compacti. On fields of moduli of curves pierre dbbes and michel emsalem department of mathematics, universitd lille, 59655 villeneuve dascq cedex, france email.
The moduli space of curves and its tautological ring, volume. The data at various temperatures should be shifted with respected to log of time until the curves merge in to a single smooth function. Download fulltext pdf nodal curves with general moduli on k3 surfaces article pdf available in communications in algebra 361. One can also enrich the problem by considering the moduli stack of genus g nodal curves with n marked points, pairwise distinct and distinct from the nodes. The rationality of certain moduli spaces of curves of genus 3. Moduli of elliptic curves peter bruin 12 september 2007 1. 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. Combining this with a separate analysis of the hyperelliptic. This is the problem underlying siegel modular form theory. Using the simplest methods, and proving as much as possible, formulas are derived for the betti numbers. Integrable systems in this section i will try to give, in a few pages, a precise idea of what an integrable system is, in the context of evolutionary hamiltonian pdes. The goal of this paper is to formulate and to begin an exploration of the enumerative geometry of the set of all curves of. We can construct moduli spaces of curves and we know their basic properties. Intuitively speaking, we can describe an elliptic curve over a scheme s as an algebraic family of elliptic curves, one for each point of s.
The moduli space of smooth algebraic curves of genus g is an algebraic variety denoted by mg. Themodulispaces of hyperelliptic curves andbinaryforms d. Jul 30, 2015 we show how to construct nonisotrivial families of supersingular k3 surfaces over rational curves using a relative form of the artintate isomorphism and twisted analogues of bridgelands results. The torelli group tg is the group of isotopy classes of di eo morphisms of a compact orientable surface of genus g that act trivially on the homology of the surface. Moduli of curves joe harris, ian morrison download. Moduli of curves 2 wecontinuetousetheconventionsandtheabuseoflanguageintroducedinproperties of stacks, section 2. We then show that the superstack of supersymmetric curves admits a coarse complex superspace, which, in this. 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. In particular, pic0 ea exists as a semiabelian algebraic space. We study the geometry of the moduli stack of vector bundles of. Equivariant cohomology of moduli spaces of genus three. From such foundations, the book goes on to show how moduli spaces of curves are constructed, to illustrate typical applications with the proofs of the brillnoether and gieseker petri theorems via limit linear series, and to survey the most important results about their geometry ranging from irreducibility and complete subvarieties to ample. An introduction to moduli spaces of curves and its. Any help or suggestions to achieve this requirement.
Pdf on moduli of pointed real curves of genus zero. Letcurves bethestackwhose category of sections over a scheme sis the category of families of curves over s. Developing master curves, binder viscosity and predicting. Intersection theory on the moduli space of curves and the matrix airy function maxim kontsevich maxplanck institut fur mathematik, gottfriedclarenstrabe 26, w5300 bonn 1, federal republic of germany received october 29, 1991. Knudsen established the existence of a projective moduli space for stable curves. The previous exercise shows that the coarse moduli scheme of smooth curves assuming it exists cannot be proper. Lange 1 introduction every smooth hyperelliptic curve of genus g. Geometric invariant theory and moduli spaces of pointed curves david swinarski ph. However, whereas universal tricanonically embedded stable curves are easy to construct via hilbert schemes, we need to do additional work in the case of generalized elliptic curves because the group law involves the possibly nonproper smooth locus.
Composite curve merging two or more curves into one youtube. Mori theory of moduli spaces of stable curves projective press. The tautological ring of the moduli space of curves. Don zagier, john harer, the euler characteristic of the moduli space of curves, inventiones mathematicae 1986 volume. Abstract the aim of this thesis is to explore the moduli of pointed real curves of genus zero. An introduction to moduli spaces of curves and their. Its appeal results not only from its beautiful mathematical structure but also from recent developments in theoretical physics, in. T ur curves by the universal property of the quotient ur, see groupoids in spaces, lemma 22. The notion of length is lost, but w e retain the notion of angle, for if v 1. Aims theaimofthisbookistoprovideaguidetoarichandfascinatings ject.
Str uctures on a surf ce w e start with t w o notions from linear algebra. The goal of this article is to motivate and describe how gromovwitten theory can and has provided tools to understand the moduli space of curves. The past decade has seen substantial progress in understanding the moduli space of curves, involving ideas, for example, from geometry algebraic, symplectic, and differential, physics, topology, and. Pdf moduli stacks of curves with a fixed dual graph. In section 1 we discuss the general problem of constructing.
Msri series volume 28, 1995 torelli groups and geometry of moduli spaces of curves richard m. By smooth curve, we mean a compact also known as proper or complete, smooth also known as nonsingular complex curve, i. Combining this vanishing with poincare duality and the long exact sequence of. We introduce the moduli space rm2k,l of pointed real curves of genus zero and give its natural strati. We study the moduli space of metric graphs that arise from tropical plane curves.
We show that two natural approaches to quantum gravity coincide. Families of curves and variation in moduli alessio del padrone and ernesto mistretta 4th july 2006 abstract in this paper we study the class of smooth complex projective varieties bsuch that any modular morphism b. Intersection theory on the moduli space of curves and the. We begin by looking again at moduli of curves of genus 1. The moduli space of curves is rigid paul hacking august 6, 2008 abstract we prove that the moduli stack m g,n of stable curves of genus g with nmarked points is rigid, that is, has no in. The tautological ring of the moduli space of stable curves m gis substantially more complicated because there are many more tautological classes than just polynomials in the kappa classes. One should nevertheless remark that git still gives stronger results about what line bundles on the moduli space are ample. Dec 30, 20 merging two or more curves into a single one sometimes proves very useful in reducing features in your model tree. Git constructions of moduli spaces of stable curves and maps ian morrison abstract. Geometry of moduli stacks of k,lstable vector bundles over algebraic curves o. Master curves are constructed using the principle of timetemperature superposition. An elliptic curve considered over the complex numbers is a pair e,o where e is a curve, which topologically is a torus, and o is a point on e. If anyone can please suggest any third part party dll which can merge all the pdfs and also convert merged pdf into png image in.
Mg for the moduli space of smooth complete curves of genus gdm69. 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. A moduli space mp for the above problem consists of a space in. Pdf nodal curves with general moduli on k3 surfaces. The bubbling phenomenon is the primary cause of noncompactness of moduli space of jholomorphic curves, and when.
Trying to combine 2 curves to 1 object autodesk community. Geometry of algebraic curves university of chicago. 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. Such marked curves are said to be stable if the subgroup of curve automorphisms which fix the marked points is finite. S2 as what we currently consider, it is the only cause. Themodulispaces of hyperelliptic curves andbinaryforms. This work is a comprehensive treatment of recent developments in the study of elliptic curves and their moduli spaces. We consider the stack of stable curves of genus g with a given dual graph and we give an explicit desingularization of its closure in the moduli stack of stable curves. They also study stable relative maps which combine features of. Trying to combine 2 curves to 1 object im creating foot controls to rig my character but im having a problem with the controller ive created.
This is true even for moduli spaces traditionally under the umbrella of git like vector bundles fa. There are far fewer such graphs than tropicalizations of classical plane curves. Di erential geometry and geometric group theory the moduli space of tropical curves is a tool to study the outer automorphism group of the free group fn, in a similar fashion to how teichmueller space is used to study the mapping class group. The data at various temperatures should be shifted with respected to log of time until the curves merge. The cohomology of the moduli space of pointed genus zero curves. Nodal curves with general moduli on k3 surfaces flaminio flamini 1, andreas leopold knutsen 2, gianluca acienzap 3 and edoardo sernesi 4 abstract. Compactifying moduli spaces american mathematical society. 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.
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