Compactifications of the generalized Jacobian variety

Oda, Tadao; Seshadri, C. S.

doi:10.1090/s0002-9947-1979-0536936-4

Cited by 136 publications

(131 citation statements)

References 21 publications

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“…For sheaves on curves the main reference is Seshadri [Ses82], for a narrower case of irreducible curves (where the moduli space is fine) the references are works [D'S79, AIK77, AK80, AK79] of D'Souza, Altman, Iarrobino, Kleiman and Altman. For another special case, of nodal but possibly reducible curves, it is Oda-Seshadri [OS79]. For a more general case, of semistable coherent sheaves on arbitrary projective families, see [Sim94].…”

Section: 1mentioning

confidence: 99%

“…Oda and Seshadri [OS79] constructed a number of compactified jacobians Jac φ of nodal curves. These jacobians are exactly the same as the projective schemes Jac d,L of the previous section.…”

Section: 1mentioning

confidence: 99%

“…(1) The semistability condition in [OS79] is formulated not in terms of subsheaves but as φ-semistability, depending on a parameter φ ∈ ∂C 1 (Γ, R), where Γ is the dual graph of the curve X.…”

Section: 1mentioning

confidence: 99%

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Compactiﬁed Jacobians and Torelli Map

Alexeev¹

2004

Publ. Res. Inst. Math. Sci.

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We compare several constructions of compactified jacobians -using semistable sheaves, semistable projective curves, degenerations of abelian varieties, and combinatorics of cell decompositions -and show that they are equivalent. We give a detailed description of the "canonical compactified jacobian" in degree g − 1. Finally, we explain how Kapranov's compactification of configuration spaces can be understood as a toric analog of the extended Torelli map.

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Section: 1mentioning

confidence: 99%

Section: 1mentioning

confidence: 99%

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Compactiﬁed Jacobians and Torelli Map

Alexeev¹

2004

Publ. Res. Inst. Math. Sci.

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“…Although in the literature there exist several different constructions of compactified jacobians, recent work of V. Alexeev shows that in case d = g − 1, there exists a "canonical" one. More precisely, in [Al04] the compactifications of [OS79], [S94] and [C94] are shown to be isomorphic if d = g − 1, and to behave consistently with the degeneration theory of principally polarized abelian varieties, as we will explain below.…”

Section: Introductionmentioning

confidence: 93%

“…The only novelty is that we use line bundles on the partial normalizations of X, rather than torsion free sheaves on X (as in [AK80], [OS79], [S94] among others) or line bundles on the blow-ups of X (as in [C94]). …”

Section: Althoughmentioning

confidence: 99%

Notation and Conventions

Cairoli

Dalang

1996

Sequential Stochastic Optimization

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Counting Geometric Points on Families of Abelian Varieties

Call

1994

Mathematische Nachrichten

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The fundamental problem in Diophantine geometry is, given a variety V defined over a number field k, to describe the set of k-rational points on V In this paper we consider a family of abelian varieties z : Y --f T parametrized by a curve T In particular, our results apply to the case where Vis an elliptic surface. A point lying on the image of a section of this family is called a geometric point. We count the number of geometric points on Vas a function of their height in two ways: first, by using canonical heights on the special fibers of Yand, second, by choosing a Weil height on a completion of K Our strategy is to estimate the height of a geometric point in terms of the height of the section on which it lies (considered as a point on the generic fiber of n : V + T ) and the height of the point on the base curve above which it lies. To derive the key estimates needed for our counting results, we extend the theorems of TATE [20], LANG [Ill and GREEN [7], which describe how the canonical height of a geometric point varies in terms of a height function on the base.We will now outline the contents of the paper in more detail. Let T be a smooth projective curve defined over the number field k. Put K = k ( T ) and let A be an abelian variety over the function field K. Let d / T b e the NCron model of A over K, and let r denote the group of sections of d / T Recall that, by the definition of the Ntron model, every point P A E A(K) extends to a section P: T --f d, so r is naturally isomorphic to A(K). By the Lang-Neron theorem (Cf. [ll], Chapter 6), if (B, z) is the K/k-trace of A, then the quotient group A(K)/zB(k) is finitely generated. Thus we will fix a subgroup A of r such that A injects into A(K)/zB(k). Note that if dim A = 1 and the j-invariant of A is transcendental over k, then .rB(k) = 0 and we may simply take A = r.The image of each section P E A is a curve on d whose points are, by definition, geometric.Taking all the sections in A, we define the set of k-rational geometric points on d determined by A to be Ageo(k) = {Pt E d ( k ) : P E A and t E T(k)} .

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Compactifications of the generalized Jacobian variety

Cited by 136 publications

References 21 publications

Compactiﬁed Jacobians and Torelli Map

Compactiﬁed Jacobians and Torelli Map

Notation and Conventions

Counting Geometric Points on Families of Abelian Varieties

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