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. 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 day The Moduli Space of Curves SpringerLink Skip to main content Skip to table of contents. 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. It is thus a special case of a moduli space. The moduli space of curves has proven itself a central object in geometry. The past decade has seen substantial progress in understanding the moduli space of curves, in- volvingideas,forexample,fromgeometryalgebraic,symplectic,anddifferential,physics, topology, and combinatorics. Many of the new ideas are related to the tautological ring of the moduli space, a subring of the cohomology or Chow ring.

In mathematics, in particular algebraic geometry, a moduli space is a geometric space usually a scheme or an algebraic stack whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such spaces frequently arise as solutions to classification problems: If one can show that a collection of interesting objects e.g., the smooth algebraic. By “moduli space of curves” we mean a “parameter space for curves”. As a ﬁrst approximation, we mean the set of curves, but we want to endow this set with further structure ideally that of a manifold, or even of a smooth complex variety. This structure should be given by nature, not arbitrarily deﬁned. The workshop will focus on recent progress in the understanding of the cohomology and cycle theory of the moduli space of curves. Topics include cohomology calculation, Gorenstein questions, tautological relations, cycles on the universal Jacobian, Hurwitz loci, and r-spin curves. [EH5] D. Eisenbud and J. Harris. The Kodaira dimension of the moduli space of curves of genus 23. Invent. Math. 901987, 359387. [EH6] D. Eisenbud and J. Harris. When rami cation points meet. Invent. Math. 871987, 485493. [Fab] C. Faber. A conjectural description of the tautological ring of the moduli space of curves. In Moduli of curves. The moduli space is therefore a dense a ne open subset of the pro-jective irreducible GIT quotient P==SLn 1. 2. Quadratic line complexes Mumford’s trick can be used to construct a quasiprojective moduli space for smooth sections Xof any Y ˆPn by degree-dhypersurfaces whenever AutY is reductive, the embedding X,!Y is canonical.

Preface Aims Theaimofthisbookistoprovideaguidetoarichandfascinatingsub-ject:algebraiccurves,andhowtheyvaryinfamilies.Therevolution. The key feature of the space RP1, considered as the moduli space of lines in the plane, is that it captures the ways in which lines can modulate, or vary contin-uously in families. But when do families of lines arise? A good example is provided by the following construc-tion. Whenever we have a continuous curve. Because C is the coarse moduli space, jBb=jBbif and only if the fibers π−1b and π−1b are isomorphic elliptic curves. Moti-vated by the concept of classifying space in topology, we require that a moduli space have a universal family. This means that if C were the moduli space of elliptic curves, there would exist. This space is a compactification of the moduli space of smooth, projective curves, constructed by Deligne and Mumford. It has played an important role in algebraic geometry and has connections to many different parts of mathematics e.g. graph theory, the theory. Shing-Tung Yau Geometry of the Moduli Space of Curves The moduli space has also played important role in physics. For examples, the Witten conjecture, proved by Kontsevich, states that the intersection numbers of the ψ-classes are governed by the KdV hierarchy. Other elegant proofs are given by Kim-Liu, Mirzakhani and Okounkov-Pandhripande.

elliptic curves over Ki.e. the equation de ning the curve has coe cients in K up to isomorphism. We consider curves E 1;E 2 over Kto be isomorphic if there is a map ’: E 1!E 2 de ned over K meaning the coe cients of ’are in K. This is an example of a moduli problem. We begin by studying EllC. Elliptic curves over C have a particularly. Basic concepts in moduli theory. Let $ S $ be a scheme a complex or algebraic space. A family of objects parametrized by the scheme $ S $ or, as is often said, "scheme over Sover S" or "scheme with basis Swith basis S" is a set of objects$$ \ X _s: s \in S, X _s \in A \, $$ equipped with an additional structure compatible with the structure of the base $ S $. Jul 07, 2014 · J. Harris, D. Mumford, "On the Kodaira dimension of the moduli space of curves" Invent. Math., 67 1982 pp. 23–86 MR0664324 Zbl 0506.14016 [a4] S. Mori, S. Mukai, "Uniruledness of the moduli space of curves of genus 11" M. Reynard ed. T. Shioda ed., Algebraic geometry, Lect. notes in math., 1016, Springer 1983 pp. 334–353 MR0726433.

8 RICHARD HAIN AND MAKOTO MATSUMOTO 4. Moduli Spaces of Elliptic Curves Suppose that r and n are non-negative integers satisfying rn > 0. Denote the moduli stack over SpecZ of smooth elliptic curves with n marked points and r non-zero tangent vectors by M1;n⃗r.Here the n marked points and the anchor points of the r tangent vectors are distinct. Taking each tangent vector to its anchor. A moduli space is the orbit space of a group action, or more gen- erally, of a groupoid. Various theories Floer homology, quan- tum cohomology, etc deﬁne topological invariants using them. Often the objects are stable, meaning that the isotropy auto- morphism groups are ﬁnite. Dec 09, 2008 · Download PDF Abstract: These informal notes are an expanded version of lectures on the moduli space of elliptic curves given at Zhejiang University in July, 2008. 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.

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