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Syzygies of abelian varieties
 J. Amer. Math. Soc
"... This is the first in a series of papers meant to introduce a notion of regularity on abelian varieties and more general irregular varieties. This notion, called Mukai regularity, is based on Mukai’s concept of Fourier transform, and in a very particular form (called Theta regularity) it parallels an ..."
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Cited by 39 (8 self)
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This is the first in a series of papers meant to introduce a notion of regularity on abelian varieties and more general irregular varieties. This notion, called Mukai regularity, is based on Mukai’s concept of Fourier transform, and in a very particular form (called Theta regularity) it parallels and strengthens the usual CastelnuovoMumford
Visualizing elements in the ShafarevichTate group
 Math
, 1998
"... this article, one can try go the other way: given an elliptic curve, and a Selmer class, find the explicit equations of the curve of genus 1 representing that class. There is a wealth of material which goes in the first direction (e.g., typical of such is the result of Cassels about plane diagonal c ..."
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Cited by 24 (2 self)
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this article, one can try go the other way: given an elliptic curve, and a Selmer class, find the explicit equations of the curve of genus 1 representing that class. There is a wealth of material which goes in the first direction (e.g., typical of such is the result of Cassels about plane diagonal cubics: for nonzero constants
The Jacobian and Formal Group of a Curve of Genus 2 over an Arbitrary Ground
 Math. Proc. Cambridge Philos. Soc. 107
, 1990
"... The ability to perform practical computations on particular cases has greatly influenced the theory of elliptic curves. First, it has allowed a rich subbranch of the Mathematics of Computation to develop, devoted to elliptic curves: the search for curves of large rank, large torsion over number fie ..."
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Cited by 18 (3 self)
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The ability to perform practical computations on particular cases has greatly influenced the theory of elliptic curves. First, it has allowed a rich subbranch of the Mathematics of Computation to develop, devoted to elliptic curves: the search for curves of large rank, large torsion over number fields, and — more recently — the application of
CalabiYau threefolds and moduli of abelian surfaces
"... be the moduli space of polarized abelian surfaces with canonical level structure. Both are (possibly singular) quasiprojective threefolds, and Alev d is a finite cover of Ad. We will also denote by Ãd and Ãlev d nonsingular models of suitable compactifications of these moduli spaces. We will use in ..."
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Cited by 17 (3 self)
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be the moduli space of polarized abelian surfaces with canonical level structure. Both are (possibly singular) quasiprojective threefolds, and Alev d is a finite cover of Ad. We will also denote by Ãd and Ãlev d nonsingular models of suitable compactifications of these moduli spaces. We will use in the sequel definitions and notation as in [GP1], [GP2]; see also [Mu1], [LB] and [HKW] for basic facts concerning abelian varieties and their moduli. Throughout the paper the base field will be C. Let Ad denote the moduli space of polarized abelian surfaces of type (1,d), and let A lev d The main goal of this paper, which is a continuation of [GP1] and [GP2], is to describe birational models for moduli spaces of these types for small values of d. Since the Kodaira dimension is a birational invariant, thus independent of the chosen compactification, we can decide the uniruledness, unirationality or rationality of nonsingular models of (any) compactifications of these moduli spaces.
Hurwitz Spaces of Genus 2 Covers of an Elliptic Curve
, 2001
"... Introduction Let E be an elliptic curve over a field K of characteristic 6= 2 and let N ? 1 be an integer prime to char(K). The purpose of this paper is to study the family of genus 2 covers of E of fixed degree N , i.e. those covers f : C ! E for which C=K is a curve of genus 2 and deg(f) = N . Si ..."
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Cited by 15 (0 self)
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Introduction Let E be an elliptic curve over a field K of characteristic 6= 2 and let N ? 1 be an integer prime to char(K). The purpose of this paper is to study the family of genus 2 covers of E of fixed degree N , i.e. those covers f : C ! E for which C=K is a curve of genus 2 and deg(f) = N . Since we can (without loss of generality) restrict our attention those covers that are normalized in the sense of section 2, this investigation is essentially equivalent to the study of the set CovE=K;N (K) := ff : C ! E : g C = 2; deg(f) = N and f is normalizedg=' of isomorphism cla
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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Cited by 14 (3 self)
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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...
Regularity on abelian varieties. II. Basic results on linear series and defining equations
 J. Algebraic Geom
"... This paper is mainly concerned with applying the theory of Mukai regularity (or Mregularity) introduced in [PP] to the study of linear series given by multiples of ample line bundles on abelian varieties. We show that this regularity notion allows one to define a new invariant of a line bundle, cal ..."
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Cited by 14 (5 self)
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This paper is mainly concerned with applying the theory of Mukai regularity (or Mregularity) introduced in [PP] to the study of linear series given by multiples of ample line bundles on abelian varieties. We show that this regularity notion allows one to define a new invariant of a line bundle, called Mregularity index, which will be seen to roughly
Equations of (1; d)polarized abelian surfaces
 Math. Ann
, 1998
"... §0. Introduction. In this paper, we study the equations of projectively embedded abelian surfaces with a polarization of type (1, d). Classical results say that given an ample line bundle L on an abelian surface A, the line bundle L ⊗n is very ample for n ≥ 3, and furthermore, in case n is even and ..."
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Cited by 14 (6 self)
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§0. Introduction. In this paper, we study the equations of projectively embedded abelian surfaces with a polarization of type (1, d). Classical results say that given an ample line bundle L on an abelian surface A, the line bundle L ⊗n is very ample for n ≥ 3, and furthermore, in case n is even and n ≥ 4, the generators of the homogeneous ideal IA of the embedding of A
Periodindex problems in WCgroups II: abelian varieties
"... We study the relationship between the period and the index of a principal homogeneous space over an abelian variety, obtaining results which, in particular, generalize work of Cassels and Lichtenbaum on curves of genus one. In addition, we show that the ptorsion in the ShafarevichTate group of a ..."
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Cited by 10 (9 self)
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We study the relationship between the period and the index of a principal homogeneous space over an abelian variety, obtaining results which, in particular, generalize work of Cassels and Lichtenbaum on curves of genus one. In addition, we show that the ptorsion in the ShafarevichTate group of a fixed abelian variety over a number field k grows arbitarily large when considered over field extensions l/k of bounded degree. Essential use is made of an abelian variety version of O’Neil’s periodindex obstruction.