Moduli of theta-characteristics via Nikulin surfaces

We study moduli spaces of K3 surfaces endowed with a Nikulin involution and their image in the moduli space R _g of Prym curves of genus g. We observe a striking analogy with Mukai’s well-known work on ordinary K3 surfaces. Many of Mukai’s results have a very precise Prym-Nikulin analogue, for instance a general Prym curve from R _g is a section of a Nikulin surface if and only if g ≤ 7 and g ≠ 6. Furthermore, R ₇ has the structure of a fibre space over the corresponding moduli space of polarized Nikulin surfaces. We then use these results to study the geometry of the moduli space of even spin curves, with special emphasis on the transition case of which is a 21-dimensional Calabi-Yau variety.     

On the moduli of surfaces admitting genus 2 fibrations

We investigate the structure of the components of the moduli space of surfaces of general type, which parametrize surfaces admitting nonsmooth genus 2 fibrations of nonalbanese type, over curves of genusg _b≥2.     

p -SOLVABILITY AND A GENERALIZATION OF PRIME GRAPHS OF FINITE GROUPS

Abstract

The real Torelli mapping, from the moduli space of real curves of genus g to the moduli space of g-dimensional real principally polarized abelian varieties, sends a real curve into its real Jacobian. The real Schottky problem is to describe its image. The results contained in the present paper concern hyperelliptic real curves and in particular real curves of genus 2. We exhibit also some counterexamples for the non-hyperelliptic case.     

Surfaces with pg = 2, K2 = 3 and a pencil of curves of genus 2

We classify minimal smooth surfaces of general type with K ² = 3, p _g = 2 which admit a fibration of curves of genus 2.We prove that they form an irreducible set of dimension 22 in their moduli space.      

Pants Decompositions of Random Surfaces

Our goal is to show, in two different contexts, that “random” surfaces have large pants decompositions. First we show that there are hyperbolic surfaces of genus g for which any pants decomposition requires curves of total length at least g ^7/6−ε . Moreover, we prove that this bound holds for most metrics in the moduli space of hyperbolic metrics equipped with the Weil–Petersson volume form. We then consider surfaces obtained by randomly gluing euclidean triangles (with unit side length) together and show that these surfaces have the same property.     

Riemann surfaces of genus g with an automorphism of order p prime and p > g

The present work completes the classification of the compact Riemann surfaces of genus g with an analytic automorphism of order p (prime number) and p > g. More precisely, we construct a parameterization space for them, we compute their groups of uniformization and we compute their full automorphism groups. Also, we give affine equations in for special cases and some implications on the components of the singular locus of the moduli space of smooth curves of genus g.     

$\Omega$-admissible theory II. Deligne pairings over moduli spaces of punctured Riemann surfaces

In Part I, Deligne-Riemann-Roch isometry is generalized for punctured Riemann surfaces equipped with quasi-hyperbolic metrics. This is achieved by proving the Mean Value Lemmas, which explicitly explain how metrized Deligne pairings for -admissible metrized line bundles depend on . In Part II, we first introduce several line bundles over Knudsen-Deligne-Mumford compactification of the moduli space (or rather the algebraic stack) of stable N-pointed algebraic curves of genus g, which are rather natural and include Weil-Petersson, Takhtajan-Zograf and logarithmic Mumford line bundles. Then we use Deligne-Riemann-Roch isomorphism and its metrized version (proved in Part I) to establish some fundamental relations among these line bundles. Finally, we compute first Chern forms of the metrized Weil-Petersson, Takhtajan-Zograf and logarithmic Mumford line bundles by using results of Wolpert and Takhtajan-Zograf, and show that the so-called Takhtajan-Zograf metric on the moduli space is algebraic. Received February 14, 2000 / Accepted August 18, 2000 / Published online February 5, 2001     

Hyperelliptic Surfaces and their Moduli

The hyperelliptic portion of the moduli space of compact Riemann surfaces of genus g2 is decomposed into a lattice of nondisjoint subvarieties corresponding precisely with the lattice of maximal g-hyperelliptic group actions (classified up to topological equivalence). The resulting stratification of the hyperelliptic moduli space exhibits regularities which depend on the parity of g and can be detected at the level of groups of order 8.     

The orbifold cohomology of moduli of genus 3 curves

In this work we study the additive orbifold cohomology of the moduli stack of smooth genus g curves. We show that this problem reduces to investigating the rational cohomology of moduli spaces of cyclic covers of curves where the genus of the covering curve is g. Then we work out the case of genus g =  3. Furthermore, we determine the part of the orbifold cohomology of the Deligne–Mumford compactification of the moduli space of genus 3 curves that comes from the Zariski closure of the inertia stack of ${\mathcal{M}3}$ .     

Real structures of Teichmüller spaces

The moduli space X^g of compact Riemann surfaces of genus g, g>1, has a canonical antiholomorphic involution. It can easily be defined in terms of complex curves: a point in X^g represented by a curve C is mapped to the point represented by the complex conjugate ¯C of C. In other words, the moduli space has a canonical real structure (cf. Andreotti and Holm [2]). The Teichmüller space has, however, several essentially distinct real structures. The purpose of this note is to describe all real structures of the Teichmüller space T(g,n) of compact Riemann surfaces of genus g punctured at n points.Work supported by the EMIL AALTONEN FOUNDATION     

The κ ring of the moduli of curves of compact type

The subalgebra of the tautological ring of the moduli of curves of compact type generated by the κ classes is studied in all genera. Relations, constructed via the virtual geometry of the moduli of stable quotients, are used to obtain minimal sets of generators. Bases and Betti numbers of the κ rings are computed. A universality property relating the higher genus κ rings to the genus 0 rings is proven using the virtual geometry of the moduli space of stable maps. The λ_g-formula for Hodge integrals arises as the simplest consequence.     

Real normalized differentials and Arbarello’s conjecture

Using meromorphic differentials with real periods, we prove Arbarello’s conjecture that any compact complex cycle of dimension g - n in the moduli space M _g of smooth algebraic curves of genus g must intersect the locus of curves having a Weierstrass point of order at most n.     

On the moduli spaces of fiber bundles of curves of genus $\geqq 2$

We determine the moduli spaces parametrizing analytic fiber bundles of curves of genus g \geqq 2g \geqq 2 over curves of genus g_b > (g + 1)/2.     

On the rational cohomology of moduli spaces of curves with level structures

We investigate low degree rational cohomology groups of smooth compactifications of moduli spaces of curves with level structures. In particular, we determine H^k([`(S)]<sub>g</sub>, \mathbb Q){H^k\left({\bar S}_{g}, {\mathbb Q}\right)} for g ≥ 2 and k ≤ 3, where [`(S)]_g{{\bar S}_{g}} denotes the moduli space of spin curves of genus g.     

Compactified Picard stacks over the moduli stack of stable curves with marked points

In this paper we give a construction of algebraic (Artin) stacks endowed with a modular map onto the moduli stack of stable curves of genus g with n marked points. The stacks we construct are smooth, irreducible and have dimension 4g−3+n, yielding a geometrically meaningful compactification of the universal Picard stack parametrizing n-pointed smooth curves together with a line bundle.     

Towards the cohomology of moduli spaces of higher genus stable maps

We prove that the orbifold desingularization of the moduli space of stable maps of genus g = 1 recently constructed by Vakil and Zinger has vanishing rational cohomology groups in odd degree k < 11. Received: 29 January 2007     

Square-tiled surfaces and rigid curves on moduli spaces

We study the algebro-geometric aspects of Teichmüller curves parameterizing square-tiled surfaces with two applications.(a) There exist infinitely many rigid curves on the moduli space of hyperelliptic curves. They span the same extremal ray of the cone of moving curves. Their union is a Zariski dense subset. Hence they yield infinitely many rigid curves with the same properties on the moduli space of stable n-pointed rational curves for even n.(b) The limit of slopes of Teichmüller curves and the sum of Lyapunov exponents for the Teichmüller geodesic flow determine each other, which yields information about the cone of effective divisors on the moduli space of curves.     

A new look at the moduli space of stable hyperelliptic curves

We carry out the log minimal model program for the moduli space ${\bar H_g}We carry out the log minimal model program for the moduli space [`(H)]<sub>g</sub>{\bar H_g} of stable hyperelliptic curves and show that certain log canonical models of [`(H)]_g{\bar H_g} are isomorphic to the proper transform of [`(H)]<sub>g</sub>{\bar H_g} in the corresponding log canonical models of [`(M)]_g{\bar M_g}. For g = 3, we retrieve the compact moduli space [`(B)]<sub>8</sub>{\bar B_{8}} of binary forms as a log canonical model, and obtain a decomposition of the natural map [`(H)]₃ ? [`(B)]<sub>8</sub>{\bar H_3 \to \bar B_{8}} into successive divisorial contractions of the boundary divisors. As a byproduct, we also obtain an isomorphism of [`(B)]₈{\bar B_8} with the GIT quotient of the Chow variety of bicanonically embedded hyperelliptic curves of genus three.     

Universal moduli spaces of surfaces with flat bundles and cobordism theory

For a compact, connected Lie group G, we study the moduli of pairs (Σ,E), where Σ is a genus g Riemann surface and E→Σ is a flat G-bundle. Varying both the Riemann surface Σ and the flat bundle leads to a moduli space , parametrizing families Riemann surfaces with flat G-bundles. We show that there is a stable range in which the homology of is independent of g. The stable range depends on the genus of the surface. We then identify the homology of this moduli space in the stable range, in terms of the homology of an explicit infinite loop space. Rationally, the stable cohomology of this moduli space is generated by the Mumford-Morita-Miller κ-classes, and the ring of characteristic classes of principal G-bundles, H^∗(BG). Equivalently, our theorem calculates the homology of the moduli space of semi-stable holomorphic bundles on Riemann surfaces.We then identify the homotopy type of the category of one-manifolds and surface cobordisms, each equipped with a flat G-bundle. Our methods combine the classical techniques of Atiyah and Bott, with the new techniques coming out of Madsen and Weiss's proof of Mumford's conjecture on the stable cohomology of the moduli space of Riemann surfaces.