Compactification of the universal Picard over the moduli of stable curves

This article provides two different, but closely related, moduli problems, which in characteristic zero provide a type of compactification of the universal Picard over the moduli of stable curves. Although neither is of finite type, both are limits of a sequence of stacks, each of which is a separated algebraic stack of finite type. We discuss relations to previous compactifications and partial compactifications, give a number of examples related to this compactification, and work out the structure of its fibres over certain fixed curves. Some applications are also discussed. Received January 5, 1998; in final form April 1, 1999 / Published online July 3, 2000     

Separable integral extensions and plus closure

We show that an excellent local domain of characteristic p has a separable big Cohen–Macaulay algebra. In the course of our work we prove that an element which is in the Frobenius closure of an ideal can be forced into the expansion of the ideal to a module-finite separable extension ring. Received: 28 May 1998 / Revised version: 30 November 1998     

Flatness and fibred powers

We prove that a complex map is flat if and only if none of its fibred powers has vertical (isolated or embedded) components. A commutative algebraic analogue of this statement follows: flatness is equivalent to torsion freeness of all tensor powers. Received: 21 August 1997 / Revised version: 6 April 1998     

Rational approximation to orthogonal bases and of the solutions of elliptic equations

It is shown that given , an orthogonal basis of can be approximated by an orthogonal basis , where has integral and have rational components, such that the angle between and is at most and the length , . This improves the length of the integral approximation due to Schmidt (1995). As an application, we improve a theorem of Kocan (1995) about the minimal size of grids in the solutions of elliptic equations. Our result fits the need in Kuo and Trudinger (1990). Received January 27, 1997 / Revised version received April 1, 1998     

Finding rational points on bielliptic genus 2 curves

We discuss a technique for trying to find all rational points on curves of the form Y ²=f ₃ X ⁶+f ₂ X ⁴+f ₁ X ²+f ₀, where the sextic has nonzero discriminant. This is a bielliptic curve of genus 2. When the rank of the Jacobian is 0 or 1, Chabauty's Theorem may be applied. However, we shall concentrate on the situation when the rank is at least 2. In this case, we shall derive an associated family of elliptic curves, defined over a number field ℚα. If each of these elliptic curves has rank less than the degree of ℚα : ℚ, then we shall describe a Chabauty-like technique which may be applied to try to find all the points (x,y) defined over ℚα) on the elliptic curves, for which x∈ℚ. This in turn allows us to find all ℚ-rational points on the original genus 2 curve. We apply this to give a solution to a problem of Diophantus (where the sextic in X is irreducible over ℚ), which simplifies the recent solution of Wetherell. We also present two examples where the sextic in X is reducible over ℚ. Received: 27 November 1998 / Revised version: 4 June 1999     

Ordinary subvarieties of codimension one

In this paper we extend the properties of ordinary points of curves [10] to ordinary closed points of one-dimensional affine reduced schemes and then to ordinary subvarieties of codimension one. Received: 20 March 1998 / Revised version: 7 July 1998     

Dense subgroups of <Emphasis Type="Italic">n</Emphasis>-dimensional Möbius groups

We will derive a new discreteness condition for n-dimensional M?bius subgroups as well as obtain some results concerning classification of such groups. We will also discuss dense subgroups of n-dimensional M?bius groups. The main result is that any dense group of an n-dimensional M?bius group contains a dense subgroup which is generated by at most n elements if . Received: 5 June 2001 / Published online: 24 February 2003 RID="*" ID="*" The research was partly supported by FNS of China, grant number 19801011     

Generalized Poincaré series for models¶of the braid arrangements

Let be the complexified Coxeter arrangement of hyperplanes of type A _{n −1} (n≥ 3). It is well known that the “minimal” projective De Concini–Procesi model of is isomorphic to the moduli space of stable n plus;1-pointed curves of genus 0. In this paper we study, from the point of view of models of arrangements, the action of the symmetric group Σ _n on the integer cohomology ring of . In fact we find a formula for the generalized Poincaré series which encodes all the information about this representation of Σ _n . This formula, which is obtained by using the elementary combinatorial properties of a ℤ-basis of and turns out to be very direct, should be compared with a more general result due to Getzler (see [5]). Received: 24 November 1997 / Revised version: 23 April 1998     

On some results of Morita and their application to questions of ampleness

We give an algebro-geometric proof of some results of Morita concerning the pull back of a cohomology class in the Jacobian fibration of a family of curves to the family itself. We apply those to the question of ampleness of line bundles on the family of curves and its symmetric products. Received: 3 April 2001; in final form: 1 October 2001/ Published online: 4 April 2002     

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     

Rational curves and ampleness properties¶of the tangent bundle of algebraic varieties

Let X be a projective manifold, a locally free ample subsheaf of the tangent bundle T _X . If and or n, we prove that . Furthermore we investigate ampleness properties of T _X on large families of curves and the relation to rational connectedness. Received: 2 July 1996     

Automorphism groups of real algebraic curves which are double covers of the real projective plane

For each integer g≥ 3 we give the complete list of groups acting as full automorphism groups of real algebraic curves of genus $g$ which are double covers of the real projective plane. Explicit polynomial equations of such curves and the formulae of their automorphisms are also given. Received: 29 April 1999 / Revised version: 26 November 1999