Moduli Space Of Genus Research Articles

A Birational Contraction of Genus 2 Tails in the Moduli Space of Genus 4 Curves I

A Birational Contraction of Genus 2 Tails in the Moduli Space of Genus 4 Curves I

The moduli space of genus four even spin curves is rational

Using the Mori theory for threefolds, we prove that the moduli space of pairs of smooth curves of genus four and theta characteristics without global sections is a rational variety.

The geometry of right-angled Artin subgroups of mapping class groups

We describe sufficient conditions which guarantee that a finite set of mapping classes generate a right-angled Artin group quasi-isometrically embedded in the mapping class group. Moreover, under these conditions, the orbit map to Teichmüller space is a quasi-isometric embedding for both of the standard metrics. As a consequence, we produce infinitely many genus h surfaces (for any h at least 2) in the moduli space of genus g surfaces (for any g at least 3) for which the universal covers are quasi-isometrically embedded in the Teichmüller space.

Topological Relations on Witten–Kontsevich and Hodge Potentials

Let Mg;n denote the moduli space of genus g stable alge- braic curves with n marked points. It carries the Mumford cohomology classesi. A homology class in H�(Mg;n) is said to be �-zero if the integral of any monomial in the �-classes vanishes on it. We show that any �-zero class implies a partial differential equation for generating se- ries for certain intersection indices on the moduli spaces. The genus homogeneous components of the Witten-Kontsevich potential, as well as of the more general Hodge potential, which include, in addition to -classes, intersection indices for �-classes, are special cases of these generating series, and the well-known partial differential equations for them are instances of our general construction. 2000 Math. Subj. Class. 14H10, 14H70.

Genus 2 Curves with Complex Multiplication

Genus 2 curves are useful in cryptography for both discrete-log based and pairing-based systems, but a method is required to compute genus 2 curves such that the Jacobian has a given number of points. Currently, all known methods involve constructing genus 2 curves with complex multiplication via computing their three Igusa class polynomials. These polynomials have rational coefficients and require extensive computation and precision to compute. Both the computation and the complexity analysis of these algorithms can be improved by a more precise understanding of the denominators of the coefficients of the polynomials. The main goal of this paper is to give a bound on the denominators of Igusa class polynomials of genus 2 curves with CM by a primitive quartic CM field. We give an overview of Igusa's results on the moduli space of genus two curves and the method to construct genus 2 curves via their Igusa invariants. We also give a complete characterization of the reduction type of a CM abelian surface, for biquadratic, cyclic, and non-Galois quartic CM fields, and for any type of prime decomposition of the prime, including ramified primes. The methods of the proof of the main result involve studying the embedding problem of the quartic CM field into certain matrix algebras over quaternions and invoking techniques from crystalline deformation theory.

Generalized Fermat curves

A closed Riemann surface S is a generalized Fermat curve of type ( k , n ) if it admits a group of automorphisms H ≅ Z k n such that the quotient O = S / H is an orbifold with signature ( 0 , n + 1 ; k , … , k ) , that is, the Riemann sphere with ( n + 1 ) conical points, all of same order k. The group H is called a generalized Fermat group of type ( k , n ) and the pair ( S , H ) is called a generalized Fermat pair of type ( k , n ) . We study some of the properties of generalized Fermat curves and, in particular, we provide simple algebraic curve realization of a generalized Fermat pair ( S , H ) in a lower-dimensional projective space than the usual canonical curve of S so that the normalizer of H in Aut ( S ) is still linear. We (partially) study the problem of the uniqueness of a generalized Fermat group on a fixed Riemann surface. It is noted that the moduli space of generalized Fermat curves of type ( p , n ) , where p is a prime, is isomorphic to the moduli space of orbifolds of signature ( 0 , n + 1 ; p , … , p ) . Some applications are: (i) an example of a pencil consisting of only non-hyperelliptic Riemann surfaces of genus g k = 1 + k 3 − 2 k 2 , for every integer k ⩾ 3 , admitting exactly three singular fibers, (ii) an injective holomorphic map ψ : C − { 0 , 1 } → M g , where M g is the moduli space of genus g ⩾ 2 (for infinitely many values of g), and (iii) a description of all complex surfaces isogenous to a product with invariants p g = q = 0 and covering group equal to Z 5 2 or Z 2 4 .

A Compactification of 3 via K3 Surfaces

S. Kondō defined a birational period map between the moduli space of genus three curves and a moduli space of polarized K3 surfaces. In this paper we give a resolution of the period map, providing a surjective morphism from a suitable compactification of 3 to the Baily-Borel compactification of a six dimensional ball quotient.

Extremal properties of the determinant of the Laplacian in the Bergman metric on the moduli space of genus two Riemann surfaces

We study extremal properties of the determinant of the Laplacian in the Bergman metric on the moduli space of compact genus two Riemann surfaces. By a combination of analytical and numerical methods we identify four non-degenerate critical points of this function and compute the signature of the Hessian at these points. The curve with the maximal number of automorphisms (the Burnside curve) turns out to be the point of the absolute maximum. Our results agree with the mass formula for virtual Euler characteristics of the moduli space. A similar analysis is performed for Bolza’s strata of symmetric Riemann surfaces of genus two.

Quantum cohomology of moduli spaces of genus zero stable curves

We investigate the (small) quantum cohomology ring of the moduli spaces \(\overline {\mathcal {M}}_{0,n}\) of stable n-pointed curves of genus 0. In particular, we determine an explicit presentation in the case n = 5 and we outline a computational approach to the case n = 6.

Heegner divisors in the moduli space of genus three curves

S. Kondo used periods of K3 surfaces to prove that the moduli space of genus three curves is birational to an arithmetic quotient of a complex 6-ball. In this paper we study Heegner divisors in the ball quotient, given by arithmetically defined hyperplane sections of the ball. We show that the corresponding loci of genus three curves are given by hyperelliptic curves, singular plane quartics and plane quartics admitting certain rational splitting curves.

Euler characteristics of Teichmüller curves in genus two

We calculate the Euler characteristics of all of the Teichmuller curves in the moduli space of genus two Riemann surfaces which are generated by holomorphic one-forms with a single double zero. These curves can all be embedded in Hilbert modular surfaces and our main result is that the Euler characteristic of a Teichmuller curve is proportional to the Euler characteristic of the Hilbert modular surface on which it lies. The idea is to use techniques from algebraic geometry to calculate the fundamental classes of these Teichmuller curves in certain compactifications of the Hilbert modular surfaces. This is done by defining meromorphic sections of line bundles over Hilbert modular surfaces which vanish along these Teichmuller curves. We apply these results to calculate the Siegel-Veech constants for counting closed billiards paths in certain L-shaped polygons. We also calculate the Lyapunov exponents of the Kontsevich-Zorich cocycle for any ergodic, SL_2(R)-invariant measure on the moduli space of Abelian differentials in genus two (previously calculated in unpublished work of Kontsevich and Zorich).

The Euler characteristic of local systems on the moduli of genus 3 hyperelliptic curves

For a partition λ ={ λ1 ≥ λ2 ≥ λ3 ≥ 0} of non-negative integers, we calculate the Euler characteristic of the local system Vλ on the moduli space of genus 3 hyperelliptic curves using a suitable stratification. For some λ of low degree, we make a guess for the motivic Euler characteristic of Vλ using counting curves over finite fields. Mathematics Subject Classification (1991): 14J15, 20B25

Genus 2 fields with degree 3 elliptic subfields

In this paper we study genus 2 function fields K with degree 3 elliptic subfields. We show that the number of Aut(K)-classes of such subfields of K is 0,1,2, or 4. Also we compute an equation for the locus of such K in the moduli space of genus 2 curves.

The modular geometry of random Regge triangulations

We show that the introduction of triangulations with variable connectivity and fluctuating edge lengths (random Regge triangulations) allows for a relatively simple and direct analysis of the modular properties of two-dimensional simplicial quantum gravity. In particular, we discuss in detail an explicit bijection between the space of possible random Regge triangulations (of given genus g and with N0 vertices) and a suitable decorated version of the (compactified) moduli space of genus g Riemann surfaces with N0 punctures \U0001d510g,N0. Such an analysis allows us to associate a Weil–Petersson metric with the set of random Regge triangulations and prove that the corresponding volume provides the dynamical triangulation partition function for pure gravity.

SCHOTTKY UNIFORMIZATIONS AND RIEMANN MATRICES OF MAXIMAL SYMMETRIC RIEMANN SURFACES OF GENUS 5*

In this note we consider pairs (S, τ ), where S is a closed Riemann surface of genus five and τ : S → S is some anticonformal involution with fixed points so that K(S, τ ) = {h ∈ Aut±(S) : hτ = τh} has the maximal order 96 and S/τ is orientable. We observe that there are exactly two topologically different choices for τ . They give non-isomorphic groups K(S, τ ), each one acting topologically rigid on the respective surface S. These two cases give then two (connect) real algebraic sets of real dimension one in the moduli space of genus 5. In this note we describe these components by classical Schottky groups and with the help of these uniformizations we compute their Riemann matrices.

Hurwitz numbers and intersections on moduli spaces of curves

This article is an extended version of preprint math.AG/9902104. We find an explicit formula for the number of topologically different ramified coverings of a sphere by a genus g surface with only one complicated branching point in terms of Hodge integrals over the moduli space of genus g curves with marked points.

On Hurwitz numbers and Hodge integrals

In this paper we find an explicit formula for the number of topologically different ramified coverings C → CP 1 ( C is a compact Riemann surface of genus g) with only one complicated branching point in terms of Hodge integrals over the moduli space of genus g curves with marked points.

On the Picard group of the moduli scheme of stable curves in positive characteristic

Here we prove that the moduli space of genus g stable curves in characteristic p > 0 has (up to torsion) the same Betti numbers for the étale topology as the characteristic 0 moduli space and if p > 84( g — 1 the same Picard group (up to torsion).

A supergeometric interpretation of vertex operator superalgebras

Conformal field theory (or more specifically, string theory) and related theories (cf. [BPZ], [FS], [V], and [S]) are the most promising attempts at developing a physical theory that combines all fundamental interactions of particles, including gravity. The geometry of this theory extends the use of Feynman diagrams, describing the interactions of point particles whose propagation in time sweeps out a line in space-time, to one-dimensional “particles” (strings) whose propagation in time sweeps out a two-dimensional surface. For genus zero holomorphic conformal field theory, algebraically, these interactions can be described by products of vertex operators or more precisely, by means of vertex operator algebras (cf. [Bo] and [FLM]). However, until 1990 a rigorous mathematical interpretation of the geometry and algebra involved in the “sewing” together of different particle interactions, incorporating the analysis of general analytic coordinates, had not been realized. In [H1] and [H2], motivated by the geometric notions arising in conformal field theory, Huang gives a precise geometric interpretation of the notion of vertex operator algebra by considering the geometric structure consisting of the moduli space of genus zero Riemann surfaces with punctures and local coordinates vanishing at the punctures, modulo conformal equivalence, together with the operation of sewing two such surfaces, defined by cutting discs around one puncture from each sphere and appropriately identifying the boundaries. Important aspects of this geometric structure are the concrete realization of the moduli space in terms of exponentials of a representation of the Virasoro algebra and a precise analysis of sewing using these resulting exponentials. Using this geometric structure, Huang then introduces the notion of geometric vertex operator algebra with central charge c ∈ C, and proves that the category of geometric vertex operator algebras is isomorphic to the category of vertex operator algebras. In [F], Friedan describes the extension of the physical model of conformal field theory to that of superconformal field theory and the notion of a superstring whose propagation in time sweeps out a supersurface. Whereas conformal field theory attempts to describe the interactions of bosons, superconformal field theory attempts to describe the interactions of boson-fermion pairs. This, in particular, requires an operator D such that D 2 = @ @z . Such an operator arises naturally in supergeometry. In [BMS], Beilinson, Manin and Schechtman study some aspects of superconformal symmetry, i.e., the Neveu-Schwarz algebra, from the viewpoint of algebraic geometry. In this work, we will take a differential geometric approach, extending Huang’s geometric interpretation of vertex operator algebras to a supergeometric interpretation of vertex operator superalgebras. Within the framework of supergeometry (cf. [D], [R] and [CR]) and motivated by superconformal field theory, we define the moduli space of super-Riemann surfaces with genus zero “body”, punctures, and local superconformal coordinates vanishing at the punctures, modulo superconformal equivalence. We announce the result that any local superconformal coordinates can be expressed in terms of exponentials of certain superderivations, and that these superderivations give a representation of the Neveu-Schwarz algebra with zero central charge. We define a

The moduli space of genus four double covers of elliptic curves is rational

The moduli space of genus four double covers of elliptic curves is rational