Characterizing Jacobians via flexes of the Kummer Variety (original) (raw)
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Journal für die reine und angewandte Mathematik (Crelles Journal), 2021
We give completely algebro-geometric proofs of a theorem by T. Shiota, and of a theorem by I. Krichever, characterizing Jacobians of algebraic curves among all irreducible principally polarized abelian varieties. Shiota’s characterization is given in terms of the KP equation. Krichever’s characterization is given in terms of trisecant lines to the Kummer variety. Here we treat the case of flexes and degenerate trisecants. The basic tool we use is a theorem we prove asserting that the base locus of the linear system associated to an effective line bundle on an abelian variety is reduced. This result allows us to remove all the extra assumptions that were introduced in the theorems by the first author, C. De Concini, G.Marini, and O. Debarre, in order to achieve algebro-geometric proofs of the results above.
Abelian varieties isogenous to a Jacobian
Annals of Mathematics, 2012
We define a notion of Weyl CM points in the moduli space Ag,1 of g-dimensional principally polarized abelian varieties and show that the André-Oort conjecture (or the GRH) implies the following statement: for any closed subvariety X Ag,1 over Q a , there exists a Weyl special point [(B, µ)] ∈ Ag,1(Q a) such that B is not isogenous to the abelian variety A underlying any point [(A, λ)] ∈ X. The title refers to the case when g ≥ 4 and X is the Torelli locus; in this case Tsimerman has proved the statement unconditionally. The notion of Weyl special points is generalized to the context of Shimura varieties, and we prove a corresponding conditional statement with the ambient space Ag,1 replaced by a general Shimura variety. 5 The adjective "compact" refers to the generalized Jacobian of the curve.
Products of Jacobians as Prym-Tyurin varieties
Geometriae Dedicata, 2009
Let X 1 ,. .. , X m denote smooth projective curves of genus g i ≥ 2 over an algebraically closed field of characteristic 0 and let n denote any integer at least equal to 1 + max m i=1 g i. We show that the product JX 1 × • • • × JX m of the corresponding Jacobian varieties admits the structure of a Prym-Tyurin variety of exponent n m−1. This exponent is considerably smaller than the exponent of the structure of a Prym-Tyurin variety known to exist for an arbitrary principally polarized abelian variety. Moreover it is given by explicit correspondences.
Decomposable Jacobian Varieties in New Genera
2016
We present a new technique to study Jacobian variety decompositions using subgroups of the automorphism group of the curve and the corresponding intermediate covers. In particular, this new method allows us to produce many new examples of genera for which there is a curve with completely decomposable Jacobian. These examples greatly extend the list given by Ekedahl and Serre of genera containing such curves, and provide more evidence for a positive answer to two questions they asked. Additionally, we produce new examples of families of curves, all of which have completely decomposable Jacobian varieties. These families relate to questions about special subvarieties in the moduli space of principally polarized abelian varieties.
Abel–Prym Maps for Isotypical Components of Jacobians
Bulletin of the Brazilian Mathematical Society, New Series
Let C be a smooth non-rational projective curve over the complex field C. If A is an abelian subvariety of the Jacobian J (C), we consider the Abel-Prym map ϕ A : C → A defined as the composition of the Abel map of C with the norm map of A. The goal of this work is to investigate the degree of the map ϕ A in the case where A is one of the components of an isotypical decomposition of J (C). In this case we obtain a lower bound for deg(ϕ A) and, under some hypotheses, also an upper bound. We then apply the results obtained to compute degrees of Abel-Prym maps in a few examples. In particular, these examples show that both bounds are sharp.
Characterization of abelian varieties
Inventiones Mathematicae, 2001
We prove that any smooth complex projective variety X with plurigenera P 1 (X) = P 2 (X) = 1 and irregularity q(X) = dim(X) is birational to an abelian variety.
Curves of Genus 2 with (N, N) Decomposable Jacobians
Journal of Symbolic Computation, 2001
Let C be a curve of genus 2 and ψ1 : C −→ E1 a map of degree n, from C to an elliptic curve E1, both curves defined over C. This map induces a degree n map φ1 : P 1 −→ P 1 which we call a Frey-Kani covering. We determine all possible ramifications for φ1. If ψ1 : C −→ E1 is maximal then there exists a maximal map ψ2 : C −→ E2, of degree n, to some elliptic curve E2 such that there is an isogeny of degree n 2 from the Jacobian JC to E1 × E2. We say that JC is (n, n)-decomposable. If the degree n is odd the pair (ψ2, E2) is canonically determined. For n = 3, 5, and 7, we give arithmetic examples of curves whose Jacobians are (n, n)-decomposable.