Moduli of Abelian Varieties, Vinberg θ-Groups, and Free Resolutions

Gruson, Laurent; Sam, Steven V; Weyman, Jerzy

doi:10.1007/978-1-4614-5292-8_13

Cited by 19 publications

(44 citation statements)

References 42 publications

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“…As explained in [GSW,§2.6], this action has unique orbits of codimensions 1, 4, and 7, respectively. We denote the closures of these orbits by O 1 , O 4 , O 7 , to indicate their codimension in 3 C 7 .…”

Section: Equations For Universal Kummer Threefoldsmentioning

confidence: 91%

“…The techniques in [GSW,§6.2,§6.5] show how to start with any vector in 4 A and produce the equations for a Kummer variety in the projective space of lines in A * along with the equation for its Coble quartic. If we apply these techniques to c 1 h 1 + · · · + c 7 h 7 , then we get the above formula for F τ .…”

Section: Parametrization Of the Göpel Varietymentioning

confidence: 99%

“…It was shown in [GSW,§6.2] that its closure in P 7 is precisely the Kummer threefold over c. Here the coordinates on P 7 need to be relabeled a 1 = x 000 , . .…”

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confidence: 99%

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The Universal Kummer Threefold

Ren

Sam

Schrader

et al. 2013

Experimental Mathematics

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The universal Kummer threefold is a 9-dimensional variety that represents the total space of the 6-dimensional family of Kummer threefolds in P 7 . We compute defining polynomials for three versions of this family, over the Satake hypersurface, over the Göpel variety, and over the reflection representation of type E 7 . We develop classical themes such as theta functions and Coble's quartic hypersurface using current tools from combinatorics, geometry, and commutative algebra. Symbolic and numerical computations for genus 3 moduli spaces appear alongside toric and tropical methods.

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Section: Equations For Universal Kummer Threefoldsmentioning

confidence: 91%

Section: Parametrization Of the Göpel Varietymentioning

confidence: 99%

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The Universal Kummer Threefold

Ren

Sam

Schrader

et al. 2013

Experimental Mathematics

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“…They construct rational orbits using the geometry of the intersection of two quadric hypersurfaces, and apply this to calculate the average size of the 2-Selmer groups of a certain family of hyperelliptic Jacobians. On the other hand, for some other Vinberg representations the work of Gruson, Sam and Weyman [Gruson et al 2013] gives a relation between the geometric invariant theory and the geometry of the Jacobians of our algebraic curves, and it seems likely that this should extend to an arithmetic relation also.…”

Section: Preliminaries: Vinberg Theory Stable Involutions Subregulamentioning

confidence: 94%

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2013

Algebra Number Theory

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We prove that limits of multiplicities associated to graded families of ideals exist under very general conditions. Most of our results hold for analytically unramified equicharacteristic local rings with perfect residue fields. We give a number of applications, including a "volume = multiplicity" formula, generalizing the formula of Lazarsfeld and Mustat ,ȃ , and a proof that the epsilon multiplicity of Ulrich and Validashti exists as a limit for ideals in rather general rings, including analytic local domains. We prove a generalization of this to generalized symbolic powers of ideals proposed by Herzog, Puthenpurakal and Verma. We also prove an asymptotic "additivity formula" for limits of multiplicities and a formula on limiting growth of valuations, which answers a question posed by the author, Kia Dalili and Olga Kashcheyeva. Our proofs are inspired by a philosophy of Okounkov for computing limits of multiplicities as the volume of a slice of an appropriate cone generated by a semigroup determined by an appropriate filtration on a family of algebraic objects.

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“…[18,19] and references therein. The fundamental locus is a Coble cubic in P 8 , whose singular locus is an Abelian TOME 67 (2017), FASCICULE 5 surface given as Jacobian of a curve of genus 2 with a (3, 3)−polarisation.…”

Section: Annales De L'institut Fouriermentioning

confidence: 99%

Fano congruences of index 3 and alternating 3-forms

Poi¹,

Faenzi²,

Mezzetti³

et al. 2017

Ann. inst. Fourier

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We study congruences of lines Xω defined by a sufficiently general choice of an alternating 3-form ω in n+1 dimensions, as Fano manifolds of index 3 and dimension n−1. These congruences include the G2-variety for n=6 and the variety of reductions of projected ℙ2×ℙ2 for n=7.\ud We compute the degree of Xω as the n-th Fine number and study the Hilbert scheme of these congruences proving that the choice of ω bijectively corresponds to Xω except when n=5. The fundamental locus of the congruence is also studied together with its singular locus: these varieties include the Coble cubic for n=8 and the Peskine variety for n=9.\ud The residual congruence Y of Xω with respect to a general linear congruence containing Xω is analysed in terms of the quadrics containing the linear span of Xω. We prove that Y is Cohen-Macaulay but non-Gorenstein in codimension 4. We also examine the fundamental locus G of Y of which we determine the singularities and the irreducible components

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Moduli of Abelian Varieties, Vinberg θ-Groups, and Free Resolutions

Cited by 19 publications

References 42 publications

The Universal Kummer Threefold

The Universal Kummer Threefold

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Fano congruences of index 3 and alternating 3-forms

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