Rational classes and divisors on curves of genus 2

We describe a method of looking for rational divisor classes on a curve of genus 2. We have an algorithm to decide if a given class of divisors of degree 3 contains a rational divisor. It is known that the shape of the kernel of Cassel’s morphism (X − T) is related to the existence of rational classes of degree 1. Our key tool is the dual Kummer surface.V. G. L. Neumann supported by CNPq, Brazil     

Infinitesimal Torelli theorem for surfaces of general type with certain invariants

We prove the infinitesimal Torelli theorem for general minimal complex surfaces X's with the first Chern number 3, geometric genus 1, and irregularity 0 which have non-trivial 3-torsion divisors. We also show that the coarse moduli space for surfaces with the invariants as above is a 14-dimensional unirational variety.     

Counting curves on rational surfaces

In [CH3], Caporaso and Harris derive recursive formulas counting nodal plane curves of degree d and geometric genus g in the plane (through the appropriate number of fixed general points). We rephrase their arguments in the language of maps, and extend them to other rational surfaces, and other specified intersections with a divisor. As applications, (i) we count irreducible curves on Hirzebruch surfaces in a fixed divisor class and of fixed geometric genus, (ii) we compute the higher-genus Gromov–Witten invariants of (or equivalently, counting curves of any genus and divisor class on) del Pezzo surfaces of degree at least 3. In the case of the cubic surface in (ii), we first use a result of Graber to enumeratively interpret higher-genus Gromov–Witten invariants of certain K-nef surfaces, and then apply this to a degeneration of a cubic surface. Received: 30 June 1999 / Revised version: 1 January 2000     

Real and imaginary quadratic representations of hyperelliptic function fields

A hyperelliptic function field can be always be represented as a real quadratic extension of the rational function field. If at least one of the rational prime divisors is rational over the field of constants, then it also can be represented as an imaginary quadratic extension of the rational function field. The arithmetic in the divisor class group can be realized in the second case by Cantor's algorithm. We show that in the first case one can compute in the divisor class group of the function field using reduced ideals and distances of ideals in the orders involved. Furthermore, we show how the two representations are connected and compare the computational complexity.

     

A geometric characterization of arithmetic varieties

A result of Belyi can be stated as follows. Every curve defined over a number field can be expressed as a cover of the projective line with branch locus contained in a rigid divisor. We define the notion of geometrically rigid divisors in surfaces and then show that every surface defined over a number field can be expressed as a cover of the projective plane with branch locus contained in a geometrically rigid divisor in the plane. The main result is the characterization of arithmetically defined divisors in the plane as geometrically rigid divisors in the plane.     

Finite Abelian Groups of K3 Surfaces with Smooth Quotient

The quotient space of a K3 surface by a finite group is an Enriques surface or a rational surface if it is smooth. Finite groups where the quotient space are Enriques surfaces are known. In this paper, by analyzing effective divisors on smooth rational surfaces, the author will study finite groups which act faithfully on K3 surfaces such that the quotient space are smooth. In particular, he will completely determine effective divisors on Hirzebruch surfaces such that there is a finite Abelian co...     

On the Shafarevich conjecture for genus-2 fibrations

We prove a result about fundamental group of a smooth projective surface with ample canonical divisor which admits a genus two fibration. As a corollary we prove that the universal cover of such a surface is holomorphically convex. This proves the conjecture of Shafarevich for such surfaces. This article is dedicated to Madhav V. Nori.     

Heegner divisors in the moduli space of genus three curves

S. Kondo used periods of 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'.

     

Hilbert basis of the Lipman semigroup

In this Note, we give a new method to compute the Hilbert basis of the semigroup of certain positive divisors supported on the exceptional divisor of a normal surface singularity. Our approach is purely combinatorial and enables us to avoid the long calculation of the invariants of the ring as it is presented in the work of Alt?nok and Tosun.     

Embeddings of curves

In this paper a numerical criterion for divisors on a smooth projective surface to be very ample is given. The idea is to restrict a given divisor to a sufficient number of (not necessarily, irreducible nor reduced) curfes on the surface and prove the very ampleness of the restriction. At the end we given an application to Bordiga surfaces.     

A Fourier‐Mukai approach to the enumerative geometry of principally polarized abelian surfaces

We study twisted ideal sheaves of small length on an irreducible principally polarized abelian surface $({\mathbb T},\ell )We study twisted ideal sheaves of small length on an irreducible principally polarized abelian surface $({\mathbb T},\ell )$. Using Fourier‐Mukai techniques we associate certain jumping schemes to such sheaves and completely classify such loci. We give examples of applications to the enumerative geometry of ${\mathbb T}$ and show that no smooth genus 5 curve on such a surface can contain a $g^1_3$. We also describe explicitly the singular divisors in the linear system |2?|.     

Riemann-Roch and Abel-Jacobi theory on a finite graph

It is well known that a finite graph can be viewed, in many respects, as a discrete analogue of a Riemann surface. In this paper, we pursue this analogy further in the context of linear equivalence of divisors. In particular, we formulate and prove a graph-theoretic analogue of the classical Riemann-Roch theorem. We also prove several results, analogous to classical facts about Riemann surfaces, concerning the Abel-Jacobi map from a graph to its Jacobian. As an application of our results, we characterize the existence or non-existence of a winning strategy for a certain chip-firing game played on the vertices of a graph.     

Diviseur thêta et formes différentielles

This article concerns the geometry of algebraic curves in characteristic p > 0. We study the geometric and arithmetic properties of the theta divisor Q{\Theta} associated to the vector bundle of locally exact differential forms of a curve. Among other things, we prove that, for a generic curve of genus ≥ 2, this theta divisor Q{\Theta} is always geometrically normal. We give also some results in the case where either p or the genus of the curve is small. In the last part, we apply our results on Q{\Theta} to the study of the variation of fundamental group of algebraic curves. In particular, we refine a recent result of Tamagawa on the specialization homomorphism between fundamental groups at least when the special fiber is supersingular.     

Finite Degree Holomorphic Covers of Compact Riemann Surfaces

A conjecture of Ehrenpreis states that any two compact Riemann surfaces of genus at least two have finite degree unbranched holomorphic covers that are arbitrarily close to each other in moduli space. Here we prove a weaker result where certain branched covers associated with arithmetic Riemann surfaces are allowed, and investigate the connection of our result with the original conjecture.     

The curve of ``Prym canonical' Gauss divisors on a Prym theta divisor

The Gauss linear system on the theta divisor of the Jacobian of a nonhyperelliptic curve has two striking properties:

1) the branch divisor of the Gauss map on the theta divisor is dual to the canonical model of the curve;

2) those divisors in the Gauss system parametrized by the canonical curve are reducible.

In contrast, Beauville and Debarre prove on a general Prym theta divisor of dimension all Gauss divisors are irreducible and normal. One is led to ask whether properties 1) and 2) may characterize the Gauss system of the theta divisor of a Jacobian. Since for a Prym theta divisor, the most distinguished curve in the Gauss system is the Prym canonical curve, the natural analog of the canonical curve for a Jacobian, in the present paper we analyze whether the analogs of properties 1) or 2) can ever hold for the Prym canonical curve. We note that both those properties would imply that the general Prym canonical Gauss divisor would be nonnormal. Then we find an explicit geometric model for the Prym canonical Gauss divisors and prove the following results using Beauville's singularities criterion for special subvarieties of Prym varieties:

Theorem. For all smooth doubly covered nonhyperelliptic curves of genus , the general Prym canonical Gauss divisor is normal and irreducible.

Corollary. For all smooth doubly covered nonhyperelliptic curves of genus , the Prym canonical curve is not dual to the branch divisor of the Gauss map.

     

Log canonical foliation singularities on surfaces

We give a classification of the dual graphs of the exceptional divisors on the minimal resolutions of log canonical foliation singularities on surfaces. As an application, we show the set of foliated minimal log discrepancies for foliated surface triples satisfies the ascending chain condition and a Grauert–Riemenschneider–type vanishing theorem for foliated surfaces with special log canonical foliation singularities.     

On the finite generation of the effective monoid of rational surfaces

We give a numerical criterion for ensuring the finite generation of the effective monoid of the surfaces obtained by a blowing-up of the projective plane at the supports of zero dimensional subschemes assuming that these are contained in a degenerate cubic. Furthermore, this criterion also ensures the regularity of any numerically effective divisor on these surfaces. Thus the dimension of any complete linear system is computed. On the other hand, in particular and among these surfaces, we obtain ringed rational surfaces with very large Picard numbers and with only finitely many integral curves of strictly negative self-intersection. These negative integral curves except two (−1)-curves are all contained in the support of an anticanonical divisor. Thus almost all the geometry of such surfaces is concentrated in the anticanonical class.     

Holomorphic bundles on O(− k) are algebraic

We show that holomorphic bundles on O(?k) for k>0 are algebraic. We also show that holomorphic bundles on O(?1) are trivial outside the zero section. A corollary is that bundles on the blow-up of a surface at a point are trivial on a neighborhood of the exceptional divisor minus the exceptional divisor.