Kummer Varieties and the Moduli Spaces of Abelian Varieties

Geemen, Bert van; Geer, Gerard van der

doi:10.2307/2374657

Cited by 49 publications

(77 citation statements)

References 5 publications

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“…The potential problem with the multivaluedness here stems from the fact that for example G 5 = P 5, 1 2 is the sum of square roots of products of thetas. It could well happen, and seems perhaps not quite unlikely in view of Riemann's quartic relations and Schottky-Jung identities (for an example of the Riemann quartic relation and the identities for theta constants on the Schottky locus, see the discussion of the genus 3 situation in [4]), that the product of 32 theta constants with characteristics in a vector subspace may indeed admit a holomorphic root over M g . In this case the expression above would be a natural candidate for the superstring measure.…”

Section: Further Directionsmentioning

confidence: 99%

“…Thus the form ∆ Ξ (4) [∆] must be a (possibly zero) multiple of this Schottky equation, and thus vanishes identically on M 4 , so our ansatz does produce a vanishing cosmological constant in genus 4. It seems very hard to extend a similar kind of argument to higher genus, where the slopes of effective divisors on M g and A g are not known.…”

Section: Further Directionsmentioning

confidence: 99%

“…With the new insight from Ξ 6 [∆](Ω), they started a modern program of identifying the higher genera superstring measure from factorization constraints and syzygy/asyzygy conditions [12,13]. In [13] they also proposed an ansatz for the superstring measure in genus 3, including a θ[∆] 4 factor, subject to the condition of certain linear combinations of modular forms having a square root. However, to date such a linear combination has not been found, and may not exist (see below).…”

Section: Introductionmentioning

confidence: 99%

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Superstring Scattering Amplitudes in Higher Genus

Grushevsky

2008

Commun. Math. Phys.

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Abstract. In this paper we continue the program pioneered by D'Hoker and Phong, and recently advanced by Cacciatori, Dalla Piazza, and van Geemen, of finding the chiral superstring measure by constructing modular forms satisfying certain factorization constraints. We give new expressions for their proposed ansätze in genera 2 and 3, respectively, which admit a straightforward generalization. We then propose an ansatz in genus 4 and verify that it satisfies the factorization constraints and gives a vanishing cosmological constant. We further conjecture a possible formula for the superstring amplitudes in any genus, subject to the condition that certain modular forms admit holomorphic roots.

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Section: Further Directionsmentioning

confidence: 99%

Section: Further Directionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Superstring Scattering Amplitudes in Higher Genus

Grushevsky

2008

Commun. Math. Phys.

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“…We shall show that Θ and Θ are not immersions for g > 4 and that Θ is an immersion when g is 3. This result still holds when we extend Θ to the Satake compactification of 7^3(2,4)\H 3 .…”

Section: Riccardo Salvati Mannimentioning

confidence: 52%

“…is ( It is an hypersurface of degree 16 in P cf [3], therefore it is Cohen-Macaulay. Since the subvariety of the singular points has codimension 2, it is regular in codimension 1, thus it is normal cf.…”

Section: Comparison Of Projective Varietiesmentioning

confidence: 99%

On the projective varieties associated with some subrings of the ring of Thetanullwerte

Manni

1994

Nagoya Mathematical Journal

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Let (X, L) be a principally polarized abelian variety (ppav) of dimension g such that L is a symmetric line bundle, i.e. i L -L where i is the inversion map i(x) --x. We shall denote by X[2] the two torsion points of X which are fixed by i. For any x in X [2] we have an isomorphism(1) t*(LΫ^L 2 .Here t x is the translation map.Let ί be a point of the Siegel upper half space H^ and X be the abelian varie-As symmetric line bundle L we take C* x C/(τZ* + Z*)Here e(t) stands for exp(2τπ£). Sometime, if it will be necessary, we shall writeIt is a well known fact that a basis of H (X, L ) is given by the 2 8 theta function $ n (2r, 2z) and from (1) we have theta relation We know that all these maps are injective, cf [6] and [8].Here we shall show that the injectivity of these maps extends to the Satake compactifications and we shall characterize the points of these compactifications in terms of vanishing of Thetanullwerte. Moreover we shall show that the characterization of the reducible points, obtained in this way, is ideal-theoretic. Under θ, /^(4,8)\H^ is biholomorphic onto its image, cf [6]; Θ and Θ have the same property when g is less or equal to 2, cf [5]. We shall show that Θ and Θ are not immersions for g > 4 and that Θ is an immersion when g is 3. This result still holds when we extend Θ to the Satake compactification of 7^3(2,4)\H 3 .We are grateful to G. Lupacciolu and A. Silva for helpful discussions. Satake compactificationLet R be an associative ring with the unity, then we shall denote by

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Some Properties of Second Order Theta Functions o Prym Varieties

Izadi

Pauly

2001

Math. Nachr.

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Let P ∪P ′ be the two component Prym variety associated to an étale double cover C → C of a non-hyperelliptic curve of genus g ≥ 6 and let |2Ξ 0 | and |2Ξ ′ 0 | be the linear systems of second order theta divisors on P and P ′ respectively. The component P ′ contains canonically the Prym curve C. We show that the base locus of the subseries of divisors containing C ⊂ P ′ is exactly the curve C. We also prove canonical isomorphisms between some subseries of |2Ξ 0 | and |2Ξ ′ 0 | and some subseries of second order theta divisors on the Jacobian of C.

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Kummer Varieties and the Moduli Spaces of Abelian Varieties

Cited by 49 publications

References 5 publications

Superstring Scattering Amplitudes in Higher Genus

Superstring Scattering Amplitudes in Higher Genus

On the projective varieties associated with some subrings of the ring of Thetanullwerte

Some Properties of Second Order Theta Functions o Prym Varieties

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