Covers of the projective line and the moduli space of quadratic differentials

Consider the Hurwitz space parameterizing covers of ${\mathbb{P}^1}$ branched at four points. We study its intersection with divisor classes on the moduli space of curves. As applications, we calculate the slope of Teichmüller curves parameterizing square-tiled cyclic covers. In addition, we come up with a relation among the slope of Teichmüller curves, the sum of Lyapunov exponents and the Siegel–Veech constant for the moduli space of quadratic differentials, which yields information for the effective cone of the moduli space of curves.     

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.     

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.     

Tropical hyperelliptic curves

We study the locus of tropical hyperelliptic curves inside the moduli space of tropical curves of genus g. We define a harmonic morphism of metric graphs and prove that a metric graph is hyperelliptic if and only if it admits a harmonic morphism of degree 2 to a metric tree. This generalizes the work of Baker and Norine on combinatorial graphs to the metric case. We then prove that the locus of 2-edge-connected genus g tropical hyperelliptic curves is a (2g?1)-dimensional stacky polyhedral fan whose maximal cells are in bijection with trees on g?1 vertices with maximum valence 3. Finally, we show that the Berkovich skeleton of a classical hyperelliptic plane curve satisfying a certain tropical smoothness condition is a standard ladder of genus g.     

On the monodromy of the moduli space of Calabi–Yau threefolds coming from eight planes in {\mathbb{P}^3}

It is a fundamental problem in geometry to decide which moduli spaces of polarized algebraic varieties are embedded by their period maps as Zariski open subsets of locally Hermitian symmetric domains. In the present work we prove that the moduli space of Calabi–Yau threefolds coming from eight planes in ${\mathbb{P}^3}$ does not have this property. We show furthermore that the monodromy group of a good family is Zariski dense in the corresponding symplectic group. Moreover, we study a natural sublocus which we call hyperelliptic locus, over which the variation of Hodge structures is naturally isomorphic to wedge product of a variation of Hodge structures of weight one. It turns out the hyperelliptic locus does not extend to a Shimura subvariety of type III (Siegel space) within the moduli space. Besides general Hodge theory, representation theory and computational commutative algebra, one of the proofs depends on a new result on the tensor product decomposition of complex polarized variations of Hodge structures.     

Counting isomorphism classes of pointed hyperelliptic curves of genus 4 over finite fields with even characteristic

This paper is devoted to counting the number of isomorphism classes of pointed hyperelliptic curves over finite fields. We deal with the genus 4 case and the finite fields are of even characteristics. The number of isomorphism classes is computed and the explicit formulae are given. This number can be represented as a polynomial in q of degree 7, where q is the order of the finite field. The result can be used in the classification problems and it is useful for further studies of hyperelliptic curve cryptosystems, e.g. it is of interest for research on implementing the arithmetics of curves of low genus for cryptographic purposes. It could also be of interest for point counting problems; both on moduli spaces of curves, and on finding the maximal number of points that a pointed hyperelliptic curve over a given finite field may have.     

On Moduli Spaces,Equidistribution, Estimates,and Rational Points of Algebraic Curves

We consider the moduli spaces of hyperelliptic curves, Artin–Schreier coverings, and some other families of curves of this type over fields of characteristic p. By using the Postnikov method, we obtain expressions for the Kloosterman sums. The distribution of angles of the Kloosterman sums was investigated on a computer. For small prime p, we study rational points on curves y ² = f(x). We consider the problem of the accuracy of estimates of the number of rational points of hyperelliptic curves and the existence of rational points of curves of the indicated type on the moduli spaces of these curves over a prime finite field.     

The Moduli Space of Curves, Double Hurwitz Numbers, and Faber��s Intersection Number Conjecture

We define the dimension 2g − 1 Faber-Hurwitz Chow/homology classes on the moduli space of curves, parametrizing curves expressible as branched covers of \mathbbP₁{{\mathbb{P}_1}} with given ramification over ∞ and sufficiently many fixed ramification points elsewhere. Degeneration of the target and judicious localization expresses such classes in terms of localization trees weighted by “top intersections” of tautological classes and genus 0 double Hurwitz numbers. This identity of generating series can be inverted, yielding a “combinatorialization” of top intersections of Y{\Psi} -classes. As genus 0 double Hurwitz numbers with at most 3 parts over ∞ are well understood, we obtain Faber’s Intersection Number Conjecture for up to 3 parts, and an approach to the Conjecture in general (bypassing the Virasoro Conjecture). We also recover other geometric results in a unified manner, including Looijenga’s theorem, the socle theorem for curves with rational tails, and the hyperelliptic locus in terms of κ _g–2.     

Groups of <Emphasis Type="Italic">S</Emphasis>-Units and the Problem of Periodicity of Continued Fractions in Hyperelliptic Fields

We construct a theory of periodic and quasiperiodic functional continued fractions in the field k((h)) for a linear polynomial h and in hyperelliptic fields. In addition, we establish a relationship between continued fractions in hyperelliptic fields, torsion in the Jacobians of the corresponding hyperelliptic curves, and S-units for appropriate sets S. We prove the periodicity of quasiperiodic elements of the form \(\sqrt f /d{h^s}\), where s is an integer, the polynomial f defines a hyperelliptic field, and the polynomial d is a divisor of f; such elements are important from the viewpoint of the torsion and periodicity problems. In particular, we show that the quasiperiodic element \(\sqrt f \) is periodic. We also analyze the continued fraction expansion of the key element \(\sqrt f /{h^{g + 1}}\), which defines the set of quasiperiodic elements of a hyperelliptic field.     

2-Weierstrass points of genus 3 hyperelliptic curves with extra involutions

We consider families of curves with extra automorphisms in ?₃, the moduli space of smooth hyperelliptic curves of genus g = 3. Such families of curves are explicitly determined in terms of the absolute invariants of binary octavics. For each family of positive dimension where |Aut (C)|>4, we determine the possible distributions of weights of 2-Weierstrass points.     

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.     

Explicit Mumford isomorphism for hyperelliptic curves

Using an explicit version of the Mumford isomorphism on the moduli space of hyperelliptic curves we derive a closed formula for the Arakelov-Green function of a hyperelliptic Riemann surface evaluated at its Weierstrass points.     

A stratification of the moduli space of pointed non-singular curves

We consider the moduli space of pointed non-singular curves of genus g whose Weierstrass gap sequence has the largest gap \(\ell _g\) equal to \(2g-3\). We stratify the moduli space by the sequence of osculating divisors associated to a canonically embedded curve. A monomial basis for the space of higher orders regular differentials on the curves in each stratum is constructed. Numerical conditions are given on the semigroup imposing that one of the strata is empty. Several examples are presented.     

A Combinatorial Algorithm Related to the Geometry of the Moduli Space of Pointed Curves

As pointed out in Arbarello and Cornalba (J. Alg. Geom. 5 (1996), 705–749), a theorem due to Di Francesco, Itzykson, and Zuber (see Di Francesco, Itzykson, and Zuber, Commun. Math. Phys. 151 (1993), 193–219) should yield new relations among cohomology classes of the moduli space of pointed curves. The coefficients appearing in these new relations can be determined by the algorithm we introduce in this paper.     

Vanishing thetanull and hyperelliptic curves

Let ??_g,2 be the moduli space of curves of genus g with a level‐2 structure. We prove here that there is always a non hyperelliptic element in the intersection of four thetanull divisors in ??_6,2. We prove also that for all g ≥ 3, each component of the hyperelliptic locus in ??_g,2 is a connected component of the intersection of g – 2 thetanull divisors. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The integral monodromy of hyperelliptic and trielliptic curves

We compute the and monodromy of every irreducible component of the moduli spaces of hyperelliptic and trielliptic curves. In particular, we provide a proof that the monodromy of the moduli space of hyperelliptic curves of genus g is the symplectic group . We prove that the monodromy of the moduli space of trielliptic curves with signature (r,s) is the special unitary group . Rachel Pries was partially supported by NSF grant DMS-04-00461.     

Correction: On the Severi problem in arbitrary characteristic

In this paper, we show that Severi varieties parameterizing irreducible reduced planar curves of a given degree and geometric genus are either empty or irreducible in any characteristic. Following Severi’s original idea, this gives a new proof of the irreducibility of the moduli space of smooth projective curves of a given genus in positive characteristic. It is the first proof that involves no reduction to the characteristic zero case. As a further consequence, we generalize Zariski’s theorem to positive characteristic and show that a general reduced planar curve of a given geometric genus is nodal.