Geometry of moduli stacks of k,lstable vector bundles over algebraic curves o. The arithmetic study of the moduli spaces began with jacobis fundamenta nova in 1829, and the modern theory was erected. The notion of length is lost, but w e retain the notion of angle, for if v 1. 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. We investigate the actions of a set of natural real structures. This thesis consists of a study of elliptic curves, and of elliptic surfaces. Our curves will be connected unless we especially describe them as possibly disconnected.
Aims theaimofthisbookistoprovideaguidetoarichandfascinatings ject. Unless otherwise mentioned our base scheme will be specz. The goal of this paper is to formulate and to begin an exploration of the enumerative geometry of the set of all curves of. 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. 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. The tautological ring of the moduli space of curves. Supersymmetric curves are the analogue of riemann surfaces in super geometry. 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. We introduce the moduli space rm2k,l of pointed real curves of genus zero and give its natural strati. This is the problem underlying siegel modular form theory. An introduction to the topology of the moduli space of stable. Moduli of families of curves and extremal partitions.
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 strata of rm2k,l correspond to real curves of genus zero with di. 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. A moduli space mp for the above problem consists of a space in. 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. 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.
There will be a preliminary discussion on thursday, 22. Composite curve merging two or more curves into one youtube. 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. Combining this vanishing with poincare duality and the long exact sequence of. Letcurves bethestackwhose category of sections over a scheme sis the category of families of curves over s. Thestackofcurves 0dmj thissectionisthecontinuationofquot,section15. Moduli of curves and multiple roots emre can sertoz abstract wecompactifythemodulispaceofcurvesandmultiplesquareroots of a line bundle. Themodulispaces of hyperelliptic curves andbinaryforms d. We begin by looking again at moduli of curves of genus 1. The chenruan cohomology of moduli of curves of genus 2. Pdf nodal curves with general moduli on k3 surfaces. Given that curves in projective space can become arbitrarily singular, it is an amazing fact that the moduli space of curves can be compacti.
Mumford, in towards an enumerative geometry of the moduli space of curves publishedin1983,writes. The moduli space of smooth algebraic curves of genus g is an algebraic variety denoted by mg. Combining this with a separate analysis of the hyperelliptic. 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. By smooth curve, we mean a compact also known as proper or complete, smooth also known as nonsingular complex curve, i. Intersection theory on the moduli space of curves and the.
Moduli of families of curves for conformal and quasiconformal mappings. The perturbed ricci metric that we introduced has bounded. You can just write or draw your signatures, ideas and thoughts onto the existing document. In higher dimensions, moduli of algebraic varieties are more difficult to construct and study. Thus it is natural to ask how the curvatures vary in terms of g. Str uctures on a surf ce w e start with t w o notions from linear algebra. We study the moduli space of metric graphs that arise from tropical plane curves.
Coherent sheaves on primitive multiple curves and their moduli. The moduli space of curves and its tautological ring, volume. In particular, pic0 ea exists as a semiabelian algebraic space. Developing master curves, binder viscosity and predicting. Geometry of algebraic curves university of chicago. Msri series volume 28, 1995 torelli groups and geometry of moduli spaces of curves richard m.
Modular curves september 4, 20 the rst examples of shimura varieties we encounter are the modular curves. 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. The moduli space of curves and its tautological ring ravi vakil t he moduli space of curves has provenitself a central object in geometry. Download fulltext pdf nodal curves with general moduli on k3 surfaces article pdf available in communications in algebra 361. A moduli stack of elliptic curves is a moduli stack of elliptic curves, hence a stack. The resulting moduli space is smooth and the parametrized objects have a goodmoduliinterpretation. I have multiple curves that i want to combine into 1 object, an icon to represent my chararcters foot. 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. 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. We then show that the superstack of supersymmetric curves admits a coarse complex superspace, which, in this. Equivariant cohomology of moduli spaces of genus three. The previous exercise shows that the coarse moduli scheme of smooth curves assuming it exists cannot be proper. Compactifying moduli spaces american mathematical society.
Moduli of curves 3 conversely,letxtbeafamilyofcurves. The rationality of certain moduli spaces of curves of genus 3. We introduce the moduli space of genus, pointed curves, with enough information. Mg for the moduli space of smooth complete curves of genus gdm69.
An introduction to moduli spaces of curves and their. 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. 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. Pdf on moduli of pointed real curves of genus zero.
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. Abstract the aim of this thesis is to explore the moduli of pointed real curves of genus zero. One should nevertheless remark that git still gives stronger results about what line bundles on the moduli space are ample. Moduli of curves anand deopurkar consider a reasonable class of objects in algebraic geometry.
There has been a huge amount of work on algebraic curves. Nonsimply laced cases, international mathematics research notices, vol. This note is about invariants of moduli spaces of curves. 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. 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. Moduli of curves joe harris, ian morrison download. There are far fewer such graphs than tropicalizations of classical plane curves. Such marked curves are said to be stable if the subgroup of curve automorphisms which fix the marked points is finite. U,tr determinesa2arroww sw tsatisfyingthecocyclecondition. Themodulispaces of hyperelliptic curves andbinaryforms. 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. We study the geometry of the moduli stack of vector bundles of.
Integrable systems and moduli spaces of curves 5 1. We show that two natural approaches to quantum gravity coincide. Its appeal results not only from its beautiful mathematical structure but also from recent developments in theoretical physics, in. Knudsen established the existence of a projective moduli space for stable curves. 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. Trying to combine 2 curves to 1 object autodesk community. The slope stability of the logarithmic cotangent bundle of the dm moduli spaces, chern number inequality and other properties will follow. 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. We can construct moduli spaces of curves and we know their basic properties. Combinatorial and algebrogeometric cohomology classes on.
Using the simplest methods, and proving as much as possible, formulas are derived for the betti numbers. This is true even for moduli spaces traditionally under the umbrella of git like vector bundles fa. Git constructions of moduli spaces of stable curves and maps. Its cohomology ring has a elegant structure, and mumford suggested studying the moduli space of curves in the same way. 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.
Geometric invariant theory and moduli spaces of pointed curves. S2 as what we currently consider, it is the only cause. Lange 1 introduction every smooth hyperelliptic curve of genus g. In section 1 we discuss the general problem of constructing. 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 bubbling phenomenon is the primary cause of noncompactness of moduli space of jholomorphic curves, and when. The data at various temperatures should be shifted with respected to log of time until the curves merge. The complex analytic structure and the relation to teichmuller space is further discussed in.
Combinatorial and algebrogeometric cohomology classes on the moduli spaces of curves enrico arbarello 1and maurizio cornalba,2 dedicated to the memory of claude itzykson. Any help or suggestions to achieve this requirement. Section1is an informal introduction to moduli spaces of smooth and stable curves. 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. An introduction to moduli spaces of curves and its. One can prove that e\o can be mapped isomorphically to a curve in. Don zagier, john harer, the euler characteristic of the moduli space of curves, inventiones mathematicae 1986 volume.
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. Torelli groups and geometry of moduli spaces of curves. 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. 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. The data at various temperatures should be shifted with respected to log of time until the curves merge in to a single smooth function.
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. This work is a comprehensive treatment of recent developments in the study of elliptic curves and their moduli spaces. Pdf moduli stacks of curves with a fixed dual graph. Dec 30, 20 merging two or more curves into a single one sometimes proves very useful in reducing features in your model tree. 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. An introduction to moduli spaces of curves 2 parameters.
The moduli space m g of complete nonsingular curves of genus g admits a compacti. The orbifold euler characteristic of the moduli space of curves was originally computed in. They also study stable relative maps which combine features of. 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. Master curves are constructed using the principle of timetemperature superposition. Geometric invariant theory and moduli spaces of pointed curves david swinarski ph. Finally, we are not going to talk about curves over elds that are not algebraically closed. The weilpetersson volume grows exponentially in g see also 23, in particular, a rational multiple of. The resulting master curve of the modulus, as a function of time, formed in this manner. Pdf rational curves in the moduli of supersingular k3 surfaces. Elliptic curves over schemes the notion of elliptic curves over arbitrary schemes is indispensable for the topic of moduli spaces. In this seminar, we will read selected articles of the handbook of moduli.
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. On fields of moduli of curves pierre dbbes and michel emsalem department of mathematics, universitd lille, 59655 villeneuve dascq cedex, france email. Algebraic geometery the moduli space of tropical curves can be used to explicitly describe the. Git constructions of moduli spaces of stable curves and maps ian morrison abstract. Mori theory of moduli spaces of stable curves projective press.
Moduli of elliptic curves peter bruin 12 september 2007 1. 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. T ur curves by the universal property of the quotient ur, see groupoids in spaces, lemma 22. Other good introductions to moduli spaces include 18 and 33. Chapter 1 is that if the mth hilbert point is semistable, then the curve is semistable as a curve.
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