Vector fields on smooth threefolds vanishing on complete intersections

A result of J. Wahl shows that the existence of a vector field vanishing on an ample divisor of a projective normal variety X implies that X is a cone over this divisor. If X is smooth, X will be isomorphic to the n-dimensional projective space. This paper is a first attempt to generalize Wahl's theorem to higher codimensions: Given a complex smooth projective threefold X and a vector field on X vanishing on an irreducible and reduced curve which is the scheme theoretic intersection of two ample divisors, X is isomorphic to the 3-dimensional projective space or the 3-dimensional quadric. Received: 24 April 2001     

On line bundles over real algebraic curves

Let X be a geometrically irreducible smooth projective curve defined over the real numbers. Let nX be the number of connected components of the locus of real points of X. Let x₁,…,x? be real points from ? distinct components, with ?<nX. We prove that the divisor x₁+?+x? is rigid. We also give a very simple proof of the Harnack's inequality.     

On the zeta function of divisors for projective varieties with higher rank divisor class group

Given a projective variety X defined over a finite field, the zeta function of divisors attempts to count all irreducible, codimension one subvarieties of X, each measured by their projective degree. When the dimension of X is greater than one, this is a purely p-adic function, convergent on the open unit disk. Four conjectures are expected to hold, the first of which is p-adic meromorphic continuation to all of Cp. When the divisor class group (divisors modulo linear equivalence) of X has rank one, then all four conjectures are known to be true. In this paper, we discuss the higher rank case. In particular, we prove a p-adic meromorphic continuation theorem which applies to a large class of varieties. Examples of such varieties are projective nonsingular surfaces defined over a finite field (whose effective monoid is finitely generated) and all projective toric varieties (smooth or singular).     

Anticanonical Rational Surfaces

A determination of the fixed components, base points and irregularity is made for arbitrary numerically effective divisors on any smooth projective rational surface having an effective anticanonical divisor. All of the results are proven over an algebraically closed field of arbitrary characteristic. Applications, to be treated in separate papers, include questions involving: points in good position, birational models of rational surfaces in projective space, and resolutions for 0-dimensional subschemes of defined by complete ideals.

     

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...     

Twelve points on the projective line, branched covers, and rational elliptic fibrations

The following divisors in the space of twelve points on are actually the same: the possible locus of the twelve nodal fibers in a rational elliptic fibration (i.e. a pencil of plane cubic curves); degree 12 binary forms that can be expressed as a cube plus a square; the locus of the twelve tangents to a smooth plane quartic from a general point of the plane; the branch locus of a degree 4 map from a hyperelliptic genus 3 curve to ; the branch locus of a degree 3 map from a genus 4 curve to induced by a theta-characteristic; and several more. The corresponding moduli spaces are smooth, but they are not all isomorphic; some are finite étale covers of others. We describe the web of interconnections among these spaces, and give monodromy, rationality, and Prym-related consequences. Enumerative consequences include: (i) the degree of this locus is 3762 (e.g. there are 3762 rational elliptic fibrations with nodes above 11 given general points of the base); (ii) if is a cover as in , then there are 135 different such covers branched at the same points; (iii) the general set of 12 tangent lines that arise in turn up in 120 essentially different ways. Some parts of this story are well known, and some other parts were known classically (to Zeuthen, Zariski, Coble, Mumford, and others). The unified picture is surprisingly intricate and connects many beautiful constructions, including Recillas' trigonal construction and Shioda's -Mordell-Weil lattice. Received November 3, 1999 / Published online February 5, 2001     

Free resolutions of the defining ideal of certain rational surfaces

We consider the surface obtained from the projective plane by blowing up the points of intersection of two plane curves of same degree meeting transversely. We find minimal free resolutions of the defining ideals of these surfaces embedded in projective space by the sections of a very ample divisor class. All of the results are proven over an algebraically closed field of arbitrary characteristic.     

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.     

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.

     

Arithmetic on superelliptic curves

This paper is concerned with algorithms for computing in the divisor class group of a nonsingular plane curve of the form which has only one point at infinity. Divisors are represented as ideals, and an ideal reduction algorithm based on lattice reduction is given. We obtain a unique representative for each divisor class and the algorithms for addition and reduction of divisors run in polynomial time. An algorithm is also given for solving the discrete logarithm problem when the curve is defined over a finite field.

     

Augmented base loci and restricted volumes on normal varieties

We extend to normal projective varieties defined over an arbitrary algebraically closed field a result of Ein, Lazarsfeld, Musta??, Nakamaye and Popa characterizing the augmented base locus (aka non-ample locus) of a line bundle on a smooth projective complex variety as the union of subvarieties on which the restricted volume vanishes. We also give a proof of the folklore fact that the complement of the augmented base locus is the largest open subset on which the Kodaira map defined by large and divisible multiples of the line bundle is an isomorphism.     

On computation of the greatest common divisor of several polynomials over a finite field

We propose a probabilistic algorithm to reduce computing the greatest common divisor of m polynomials over a finite field (which requires computing m−1 pairwise greatest common divisors) to computing the greatest common divisor of two polynomials over the same field.     

Slope stability and exceptional divisors of high genus

We study slope stability of smooth surfaces and its connection with exceptional divisors. We show that a surface containing an exceptional divisor with arithmetic genus at least two is slope unstable for some polarisation. In the converse direction we show that slope stability of surfaces can be tested with divisors, and prove that for surfaces with non-negative Kodaira dimension any destabilising divisor must have negative self-intersection and arithmetic genus at least two. We also prove that a destabilising divisor can never be nef, and as an application give an example of a surface that is slope stable but not K-stable. D. Panov was supported by EPSRC grant number EP/E044859/1 and J. Ross was partially supported by the National Science Foundation, Grant No. DMS-0700419.     

Étale covers of affine spaces in positive characteristicRevêtements étales des espaces affines en caractéristique positive

We prove that every projective, geometrically reduced scheme of dimension n over an infinite field k of positive characteristic admits a finite morphism over some finite radicial extension k′ of k to projective n-space, étale away from the hyperplane H at infinity, which maps a chosen Weil divisor into H and a chosen smooth geometric point of X not on the divisor to some point not in H. To cite this article: K.S. Kedlaya, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 921–926.     

On the birational geometry of

Here we are interested in birational properties of the moduli space of stable maps to the complex projective line. In particular, we determine the class of the canonical divisor in the rational Picard group and we investigate the cones of ample and effective divisors.     

Period and index of genus one curves over global fields

The period of a curve is the smallest positive degree of Galois-invariant divisor classes. The index is the smallest positive degree of rational divisors. We construct examples of genus one curves with prescribed period and index over a given global field, as long as the characteristic of the field does not divide the period.     

Branched affine and projective structures on compact Riemann surfaces

It is first established that there exist linear manifolds of branched affine structures having certain nonpolar branch divisors and simple polar divisors on an arbitrary compact Riemann surface M of genus g≤1. When ≥2, it is shown that these linear manifolds form a complex analytic vector bundle over the manifold of simple polar divisors on M. When g=1, elliptic functions are used to construct certain projective structures on M. A partial determination is made as to which of these projective structures are affine and which are not.     

Stable Maps and Branch Divisors

We construct a natural branch divisor for equidimensional projective morphisms where the domain has lci singularities and the target is nonsingular. The method involves generalizing a divisor construction of Mumford from sheaves to complexes. The construction is valid in flat families. The generalized branch divisor of a stable map to a nonsingular curve X yields a canonical morphism from the space of stable maps to a symmetric product of X. This branch morphism (together with virtual localization) is used to compute the Hurwitz numbers of covers of the projective line for all genera and degrees in terms of Hodge integrals.