To Bill Fulton on his sixtieth birthday Abstract. In this paper we study the ample cone of the moduli space Mg,n of stable n-pointed curves of genus g. Our motivating conjecture is that a divisor on Mg,n is ample iff it has positive intersection with all 1-dimensional strata (the components of the locus of curves with at least 3g+n−2 nodes). This translates into a simple conjectural description of the cone by linear inequalities, and, as all the 1-strata are rational, includes the conjecture that the Mori cone is polyhedral and generated by rational curves. Our main result is that the conjecture holds iff it holds for g = 0. More precisely, there is a natural finite map r: M0,2g+n → Mg,n whose image is the locus Rg,n of curves with all components rational. Any 1-strata either lies in Rg,n or is numerically equivalent to a family E of elliptic tails and we show that a divisor D is nef iff D ·E ≥ 0 and r ∗ (D) is nef. We also give results on contractions (i.e. morphisms with connected fibers to projective varieties) of Mg,n for g ≥ 1 showing that any fibration factors through a tautological one (given by forgetting points) and that the exceptional locus of any birational contraction is contained in the boundary. §0 Introduction and Statement of Results. The moduli space of stable curves is among the most studied objects in algebraic geometry. Nonetheless, its birational geometry remains largely a mystery and most Mori theoretic problems in the area are entirely

In this note we shall consider some arithmetic and geometric questions concerning projective varieties X with a self-map φ and an ample line bundle L such that φ ∗ (L) ∼ = L ⊗d for some d> 1, all defined over some field k. Such a situation is interesting arithmetically, if k is a number field, because using

. This survey discusses algorithms and explicit calculations for curves of genus at least 2 and their Jacobians, mainly over number fields and finite fields. Miscellaneous examples and a list of possible future projects are given at the end. 1. Introduction An enormous number of people have performed an enormous number of computations on elliptic curves, as one can see from even a perfunctory glance at [29]. A few years ago, the same could not be said for curves of higher genus, even though the theory of such curves had been developed in detail. Now, however, polynomialtime algorithms and sometimes actual programs are available for solving a wide variety of problems associated with such curves. The genus 2 case especially is becoming accessible: in light of recent work, it seems reasonable to expect that within a few years, packages will be available for doing genus 2 computations analogous to the elliptic curve computations that are currently possible in PARI, MAGMA, SIMATH, apec...

Let n be a nonzero integer and assume that a set S of positive integers has the property that xy + n is a perfect square whenever x and y are distinct elements of S. In this paper we find some upper bounds for the size of the set S. We prove that if |n | ≤ 400 then |S | ≤ 32, and if |n |> 400 then |S | < 267.81 log |n | (log log |n|) 2. The question whether there exists an absolute bound (independent on n) for |S | still remains open. 1

Abstract. In this paper, we study bounds for the number of rational points on twists C ′ of a fixed curve C over a number field K, under the condition that the group of K-rational points on the Jacobian J ′ of C ′ has rank smaller than the genus of C ′. The main result is that with some explicitly given finitely many possible exceptions, we have a bound of the form 2r + c, where r is the rank of J ′ (K) and c is a constant depending on C. For the proof, we use a refinement of the method of Chabauty-Coleman; the main new ingredient is to use it for an extension field of Kv, where v is a place of bad reduction for C ′. 1.

INTRODUCTION In Which We Are Introduced to Our Main Theorem, and the Story Begins. We work over C . 0.1. Statement. We prove the following theorem: Theorem 0.1 (Fibered power theorem). Let X ! B be a smooth family of positive dimensional varieties of general type, with B irreducible. Then there exists an integer n ? 0, a positive dimensional variety of general type W n , and a dominant rational map X n B 9 9 KW n . Specifically, let m n : X n B 9 9 KW n be the n-pointed birational-moduli map. Then for sufficiently large n, W n is a variety of general type. The latter statement will be explained in section 3. This solves "Co

2. Boundedness results for stable families 3. Uniform version of the theorem of Parshin-Arakelov

. We prove that the fibered power conjecture of Caporaso et al. (Conjecture H, [CHM], x6) together with Lang's conjecture implies the uniformity of rational points on varieties of general type, as predicted in [CHM]; a few applications on the arithmetic and geometry of curves are stated. In an opposite direction, we give counterexamples to some analogous results in positive characteristic. We show that curves that change genus can have arbitrarily many rational points; and that curves over Fp (t) can have arbitrarily many Frobenius orbits of non-constant points. Contents 1. Introduction 21 1.1. A Few Conjectures of Lang 21 1.2. The Fibered Power Conjecture 22 1.3. Summary of Results on the Implication Side 22 1.4. Summary of Results: Examples in Positive Characteristic 24 1.5. Acknowledgments 25 2. Proof of Theorem 1.5 25 2.1. Preliminaries 25 2.2. Prolongable Points 25 2.3. Proof of Theorem 1.5 26 3. A Few Refinements and Applications in Arithmetic and Geometry 26 3.1. Proof of Theor...

this paper as a simple application of the methods of [CHM].

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