24 research outputs found
Degeneration of Prym Varieties: a Computational Approach to the Indeterminacy Locus of the Prym Map and Degenerations of Cubic Threefolds
Get PDFIn this paper we will explore the extension of the Prym period map from the moduli space of admissible double covers of stable curves to the perfect cone compactification of the moduli space of principally polarized abelian varieties. We will use the insight from Casalaina-Martin, Grushevsky, Hulek, Laza, and Dutour Sikirić to understand the indeterminacy locus of this extension of the Prym map. Using computational methods we characterize the indeterminacy locus up to codimension 10 in the case where the base curves have genus 5. The last section will be devoted to an application of the Prym period map in which we construct the necessary extension data needed to classify the intermediate Jacobian of a cubic threefold with 2A1 singularity type.
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Degeneration of Prym Varieties: a Computational Approach to the Indeterminacy Locus of the Prym Map and Degenerations of Cubic Threefolds
Get PDFIn this paper we will explore the extension of the Prym period map from the moduli space of admissible double covers of stable curves to the perfect cone compactification of the moduli space of principally polarized abelian varieties. We will use the insight from Casalaina-Martin, Grushevsky, Hulek, Laza, and Dutour Sikirić to understand the indeterminacy locus of this extension of the Prym map. Using computational methods we characterize the indeterminacy locus up to codimension 10 in the case where the base curves have genus 5. The last section will be devoted to an application of the Prym period map in which we construct the necessary extension data needed to classify the intermediate Jacobian of a cubic threefold with 2A1 singularity type.
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Stable Betti numbers of (partial) toroidal compactifications of the moduli space of Abelian varieties
No full textWe present an algorithm for explicitly computing the number of generators of the stable cohomology algebra of any rationally smooth partial toroidal compactification ofA g satisfying certain additivity and finiteness properties, in terms of the combinatorics of the corresponding toric fans. In particular, the algorithm determines the stable cohomology of the matroidal partial compactification A Matrg , in terms of simple regular matroids that are irreducible with respect to the 1-sum operation, and their automorphism groups. The algorithm also applies to compute the stable Betti numbers in close to top degree for the perfect cone toroidal compactification APerf g. This suggests the existence of an algebra structure on H top−kstable (A Perfg , Q)
Extending the Prym map to toroidal compactifications of the moduli space of abelian varieties (with an appendix by Mathieu Dutour Sikiric)
Get PDFThe main purpose of this paper is to present a conceptual approach to understanding the extension of the Prym map from the space of admissible double covers of stable curves to different toroidal compactifications of the moduli space of principally polarized abelian varieties. By separating the combinatorial problems from the geometric aspects we can reduce this to the computation of certain monodromy cones. In this way we not only shed new light on the extension results of Alexeev, Birkenhake, Hulek, and Vologodsky for the second Voronoi toroidal compactification, but we also apply this to other toroidal compactifications, in particular the perfect cone compactification, for which we obtain a combinatorial characterization of the indeterminacy locus, as well as a geometric description up to codimension six, and an explicit toroidal resolution of the Prym map up to codimension four. © 2017 European Mathematical Society
Periodic triangulations of Zn
Get PDFWe consider in this work triangulations of Z n that are periodic along Z n . They generalize the triangulations obtained from Delaunay tessellations of lattices. In certain cases we impose additional restrictions on such triangulations such as regularity or invariance under central symmetry with respect to the origin; both properties hold for Delaunay tessellations of lattices. Full enumeration of such periodic triangulations is obtained for dimension at most 4 . In dimension 5 several new phenomena happen: there are centrally-symmetric triangulations that are not Delaunay, there are non-regular triangulations (it could happen in dimension 4 ) and a given simplex has a priori infinitely many possible adjacent simplices. We found 950 periodic triangulations in dimension 5 but finiteness of the whole family is unknown
On the Voronoi Conjecture for Combinatorially Voronoi Parallelohedra in Dimension 5
Get PDFIn a recent paper, Garber, Gavrilyuk, and Magazinov [Discrete Comput. Geom., 53 (2015), pp. 245--260] proposed a sufficient combinatorial condition for a parallelohedron to be affinely Voronoi. We show that this condition holds for all 5-dimensional Voronoi parallelohedra. Consequently, the Voronoi conjecture in holds if and only if every 5-dimensional parallelohedron is combinatorially Voronoi. Here, by saying that a parallelohedron is combinatorially Voronoi, we mean that is combinatorially equivalent to a Dirichlet--Voronoi polytope for some lattice , and this combinatorial equivalence is naturally translated into equivalence of the tiling by copies of with the Voronoi tiling of . We also propose a new condition which, if satisfied by a parallelohedron , is sufficient to infer that is affinely Voronoi. The condition is based on the new notion of the Venkov complex associated with a parallelohedron and cohomologies of this complex
The complete classification of five-dimensional Dirichlet–Voronoi polyhedra of translational lattices
Get PDFThis paper reports on the full classification of Dirichlet–Voronoi polyhedra and Delaunay subdivisions of five-dimensional translational lattices. A complete list is obtained of 110 244 affine types (L-types) of Delaunay subdivisions and it turns out that they are all combinatorially inequivalent, giving the same number of combinatorial types of Dirichlet–Voronoi polyhedra. Using a refinement of corresponding secondary cones, 181 394 contraction types are obtained. The paper gives details of the computer-assisted enumeration, which was verified by three independent implementations and a topological mass formula check
VORONOI COMPLEXES IN HIGHER DIMENSIONS, COHOMOLOGY OF FOR AND THE TRIVIALITY OF
No full textWe enumerate the low dimensional cells in the Voronoi cell complexes attached to the modular groups and for , using quotient sublattices techniques for and linear programming methods for higher dimensions. These enumerations allow us to compute some cohomology of these groups and prove that , providing new knowledge on the Kummer-Vandiver conjecture.IDEX UG
Space fullerenes: A computer search for new Frank-Kasper structures
No full textA Frank-Kasper structure is a 3-periodic tiling of the Euclidean space E3by tetrahedra such that the vertex figure of any vertex belongs to four specified patterns with, respectively, 20, 24, 26 and 28 faces. Frank-Kasper structures occur in the crystallography of metallic alloys and clathrates. A new computer enumeration method has been devised for obtaining Frank-Kasper structures of up to 20 cells in a reduced fundamental domain. Here, the 84 obtained structures have been compared with the known 27 physical structures and the known special constructions by Frank-Kasper-Sullivan, Shoemaker-Shoemaker, Sadoc-Mosseri and Deza-Shtogrin
