Moduli spaces of curves and representation theory (original) (raw)
Related papers
Canadian Journal of Mathematics, 2000
This paper treats the moduli space g,1(Λ) of representations of the fundamental group of a Riemann surface of genus g with one boundary component which send the loop around the boundary to an element conjugate to exp Λ, where Λ is in the fundamental alcove of a Lie algebra. We construct natural line bundles over g,1(Λ) and exhibit natural homology cycles representing the Poincaré dual of the first Chern class. We use these cycles to prove differential equations satisfied by the symplectic volumes of these spaces. Finally we give a bound on the degree of a nonvanishing element of a particular subring of the cohomology of the moduli space of stable bundles of coprime rank k and degree d.
The Picard group of the moduli of G-bundles on a curve
1996
Let G be a complex semi-simple group, and X a compact Riemann surface. The moduli space of principal G-bundles on X, and in particular the holomorphic line bundles on this space and their global sections, play an important role in the recent applications of Conformal Field Theory to algebraic geometry. In this paper we determine the Picard group of this moduli space when G is of classical or G2 type (we consider both the coarse moduli space and the moduli stack). 1991): Primary: 14H60; Secondary: 14F05, 14L30. G be the universal covering of G; for each 2 1 G, we construct a natural 'twisted' moduli stack M e G which dominates M G . (For instance if G = PGL r , it is the moduli stack of vector bundles on X of rank r and fixed determinant of degree d, with e 2id=r = .) This moduli stack carries in each case a natural line bundle D, the determinant bundle associated to the standard representation of e G. We can now state some of our results; for simplicity we only consider the adjoint groups.
1998
We complete the proof of the fact that the moduli space of rank two bundles with trivial determinant embeds into the linear system of divisors on Picg−1CPic^{g-1}CPicg−1C which are linearly equivalent to 2Theta2\Theta2Theta. The embedded tangent space at a semi-stable non-stable bundle xioplusxi−1\xi\oplus\xi^{-1}xioplusxi−1, where xi\xixi is a degree zero line bundle, is shown to consist of those divisors in ∣2Theta∣|2\Theta|∣2Theta∣ which contain Sing(Thetaxi)Sing(\Theta_{\xi})Sing(Thetaxi) where Thetaxi\Theta_{\xi}Thetaxi is the translate of Theta\ThetaTheta by xi\xixi. We also obtain geometrical results on the structure of this tangent space.
Local structure of the moduli space of vector bundles over curves
Commentarii Mathematici Helvetici, 1996
Local structure of the moduli space of vector bundles over curves Yves LASZLO (*) 0. Introduction.-Let X be a smooth, projective and connected curve (over an algebraically closed field of characteristic zero) of genus g(X) ≥ 2. Let x be a (closed) point of X and SU(r, d) the moduli space of semi-stable vector bundles on X of rank r ≥ 2 and determinant O(dx). As usual, the geometric points of SU(r, d) correspond to S-equivalence classes [E] where E is a semi-stable rank r bundle of determinant O(dx) (another semi-stable bundle F is said to be S-equivalent to F if the graded objects gr(E) and gr(F) are isomorphic). The singular locus of SU(r, d) consists exactly of the non stable points (except if r = g(X) = 2 and d = 0. In this case, SU(r, d) = P 3 [N-R1]). In particular, except in the exceptional case above, SU(r, d) is smooth if and only if r and d are relatively prime. General facts about the action of reductive groups ensure that SU(r, d) is Cohen-Macaulay [E-H], normal and that the singularities are rational [B]. The principal aim of this paper is to give additional information about the singularities, essentially the description of the completion of the local ring at a non smooth point of SU(r, d) and to compute the multiplicity and the tangent cones at those singular points [E] which are not too bad, i.e. the corresponding graded object gr(E) of [E] has only two non isomorphic stable summands (or equivalently Aut gr(E) = G m × G m). Further, we give a complete description in the rank 2 case (proposition II.2, corollary II.3 and theorem III.4). As a corollary, we get the local form of the so called Coble quartic and prove that the Kummer variety of the Jacobian of a genus 3 non hyperelliptic curve is schematically defined by 8 cubics, the partials derivatives of the Coble quartic (theorem III.6). One could also give partial information at least if Aut gr(E) is a torus, or by using results of [P], if Aut gr(E) = Gl r (k) (the latter case essentialy means that gr(E) is the trivial bundle). But it seems to be difficult and somewhat messy to calculate for instance the multiplicity. In the remaining part of the paper, we compute the multiplicity of a generalized theta divisor of SU X (2, O) at a point [L ⊕ L ∨ ] , (*) The author was partially supported by the European Science Project "Geometry of Algebraic Varieties", Contract no. SCI-0398-C(A).
On rational maps between moduli spaces of curves and of vector bundles
arXiv: Algebraic Geometry, 2009
Let SU_C(2) be the moduli space of rank 2 semistable vector bundles with trivial de terminant on a smooth complex algebraic curve C of genus g > 1, we assume C non-hyperellptic if g > 2. In this paper we construct large families of pointed rational normal curves over certain linear sections of SU_C(2). This allows us to give an interpretation of these subvarieties of SUC(2) in terms of the moduli space of curves M_{0,2g}. In fact, there exists a natural linear map SU_C(2) -> P^g with modular meaning, whose fibers are birational to M_{0,2g}, the moduli space of 2g-pointed genus zero curves. If g < 4, these modular fibers are even isomorphic to the GIT compactification M^{GIT}_{0,2g}. The families of pointed rational normal curves are recovered as the fibers of the maps that classify extensions of line bundles associated to some effective divisors.
The Picard groups of the moduli spaces of curves
Topology, 1987
the moduli space of smooth h-pointed curves of genus g over C and by &is,h its natural compactification by means of stable curves. It is known that the Picard group of M,,, is a free Abelian group on h + 1 generators when g 2 3. This is due to Harer [4, 51 (cf. the Appendix). Instead of dealing with the Picard group of the moduli space it is usually more convenient, from a technical point of view, to work with the so-called Picard group of the moduli functor (see below for a precise definition), which we shall denote by Pit (JY~,,) if we are restricting to smooth curves and by Pit #&if we are allowing singular stable curves as well. As Mumford observes in [S], Pit (Jg.J has no torsion and contains Pit (MgJ as a subgroup of finite index (a proof of this will be sketched in the Appendix). The purpose of this note is to exhibit explicit bases for Pit (&& and for Pit (d&g,h)r which is also a free Abelian group. This is done in Theorem 2 (53), of which Theorem 1 in $2 is a special case. We shall now say a couple of words about our terminology. A family of h-pointed stable curves of genus g parametrized by S is a proper flat morphism II : V + S together with disjoint sections ol,. .. , a,, having the following properties. Each fiber n-'(s) is a connected curve of genus g having only nodes as singularities and such that each of its smooth rational components contains at least three points belonging to the union of the remaining components and of the sections; moreover, for each i, Go is a smooth point of n-'(s). Following Mumford [7,8], by a line bundle on the moduli functor Jjtg,h we mean the datum of a line bundle L, (often written L,) on S for any family F = (n : %Z + S, (rl,. .. , CT,,) of h-pointed stable curves of genus g, and of an isomorphism L, s cz*(L,) for any Cartesian square of families of h-pointed stable curves; these isomorphisms are moreover required to satisfy an obvious cocycle condition. It is important to notice that we get an equivalent definition if, in the above, we restrict to families of pointed stable curves which are, near any point of the base, universal deformations for the corresponding fiber. We write Pit (Sg,h) to denote the group l Supported in part by grants from the C.N.R. and the Italian Ministry of Public Education.
On mmm--fold Holomorphic Differentials and Modular Forms
2021
Let Γ be the Fuchsian group of the first kind [7, Section 1.7, page 28]. Examples of such groups are the important modular groups such as SL2(Z) and its congruence subgroups Γ0(N), Γ1(N), and Γ(N) [7, Section 4.2]. Let H be the complex upper half-plane. The quotient Γ\H can be compactified by adding a finite number of Γ-orbits of points in R∪{∞} called cusps of Γ and we obtain a compact Riemann surface which will be denoted by RΓ. For l ≥ 1, let H l (RΓ) be the space of all holomorphic differentials on RΓ (see [5], or Section 3) in this paper). Let m ≥ 2 be an even integer. Let Sm(Γ) be the space of (holomorphic) cusp forms of weight m (see Section 2). It is well–known that S2(Γ) is naturally isomorphic to the vector space H (RΓ) (see [7, Theorem 2.3.2]). This is employed on many instances in studying various properties of modular curves (see for example [19, Chapter 6]). In this paper we study the generalization of this concept to the holomorphic differentials of higher order. For ...