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56
Towards the ample cone of ¯ Mg,n
"... To Bill Fulton on his sixtieth birthday Abstract. In this paper we study the ample cone of the moduli space Mg,n of stable npointed curves of genus g. Our motivating conjecture is that a divisor on Mg,n is ample iff it has positive intersection with all 1dimensional strata (the components of the l ..."
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To Bill Fulton on his sixtieth birthday Abstract. In this paper we study the ample cone of the moduli space Mg,n of stable npointed curves of genus g. Our motivating conjecture is that a divisor on Mg,n is ample iff it has positive intersection with all 1dimensional 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 1strata 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 1strata 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
Questions on self maps of algebraic varieties
 J. Ramanujan Math. Soc
"... In this note we shall consider some arithmetic and geometric questions concerning projective varieties X with a selfmap 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, ..."
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In this note we shall consider some arithmetic and geometric questions concerning projective varieties X with a selfmap 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
Independence of rational points on twists of a given curve, to appear
 in Compositio Math. arXiv: math.NT/0603557 School of Engineering and Science, International University Bremen, P.O.Box 750561, 28725
"... 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 Krational points on the Jacobian J ′ of C ′ has rank smaller than the genus of C ′. The main result is that with some explicitly g ..."
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Cited by 21 (12 self)
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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 Krational 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 ChabautyColeman; 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.
On the size of Diophantine mtuples
 Math. Proc. Cambridge Philos. Soc
"... 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 t ..."
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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
Computational Aspects of Curves of Genus at Least 2
 Algorithmic number theory. 5th international symposium. ANTSII
, 1996
"... . 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 per ..."
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. 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...
Lang's Conjectures, Fibered Powers, and Uniformity
 Journal of Math
, 1996
"... . 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 ..."
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Cited by 11 (5 self)
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. 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 nonconstant 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...
A High Fibered Power Of A Family Of Varieties Of General Type Dominates A Variety Of General Type  with a few diagrams and one illustration
, 1997
"... 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 ..."
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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 npointed birationalmoduli 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
On certain uniformity properties of curves over function fields
 Compositio Math
"... 2. Boundedness results for stable families 3. Uniform version of the theorem of ParshinArakelov ..."
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2. Boundedness results for stable families 3. Uniform version of the theorem of ParshinArakelov
Uniformity of Stably Integral Points on Elliptic Curves
, 1995
"... this paper as a simple application of the methods of [CHM]. ..."
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Cited by 7 (3 self)
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this paper as a simple application of the methods of [CHM].
Lattice points on circles, squares in arithmetic progressions and sumsets of squares
, 2006
"... We discuss the relationship between various additive problems concerning squares. ..."
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We discuss the relationship between various additive problems concerning squares.