65 research outputs found
Birational automorphisms of a three-dimensional double quadric with an elementary singularity
It is proved that the group of birational automorphisms of a
three-dimensional double quadric with a singular point arising from a double
point on the branch divisor is a semidirect product of the free group generated
by birational involutions of a special form and the group of regular
automorphisms. The proof is based on the method of `untwisting' maximal
singularities of linear systems.Comment: 18 page
Birational rigidity of a three-dimensional double cone
It is proved that a three-dimensional double cone is a birationally rigid
variety. We also compute the group of birational automorphisms of such a
variety. This work is based on the method of "untwisting" maximal singularities
of linear system.Comment: 20 pages; AmsLaTe
Birational geometry of algebraic varieties with a pencil of Fano cyclic covers
We prove birational rigidity of fiber spaces π: V → P 1 , the fiber of which is a Fano cyclic cover of index 1, provided it is sufficiently twisted over the base. In particular, nonrationality of these varieties is shown and their group of birational automorphisms is computed. The proof is obtained by a combination of the classical quadratic technique of the method of maximal singularities with the technique of hypertangent divisors and the connectedness principle of Shokurov and Kollár
The inverse moment problem for convex polytopes
The goal of this paper is to present a general and novel approach for the
reconstruction of any convex d-dimensional polytope P, from knowledge of its
moments. In particular, we show that the vertices of an N-vertex polytope in
R^d can be reconstructed from the knowledge of O(DN) axial moments (w.r.t. to
an unknown polynomial measure od degree D) in d+1 distinct generic directions.
Our approach is based on the collection of moment formulas due to Brion,
Lawrence, Khovanskii-Pukhikov, and Barvinok that arise in the discrete geometry
of polytopes, and what variously known as Prony's method, or Vandermonde
factorization of finite rank Hankel matrices.Comment: LaTeX2e, 24 pages including 1 appendi
Minkowski-type and Alexandrov-type theorems for polyhedral herissons
Classical H.Minkowski theorems on existence and uniqueness of convex
polyhedra with prescribed directions and areas of faces as well as the
well-known generalization of H.Minkowski uniqueness theorem due to
A.D.Alexandrov are extended to a class of nonconvex polyhedra which are called
polyhedral herissons and may be described as polyhedra with injective spherical
image.Comment: 19 pages, 8 figures, LaTeX 2.0
Schubert calculus and Gelfand-Zetlin polytopes
We describe a new approach to the Schubert calculus on complete flag
varieties using the volume polynomial associated with Gelfand-Zetlin polytopes.
This approach allows us to compute the intersection products of Schubert cycles
by intersecting faces of a polytope.Comment: 33 pages, 4 figures, introduction rewritten, Section 4 restructured,
typos correcte
Shapes of polyhedra, mixed volumes and hyperbolic geometry
We generalize to higher dimensions the Bavard–Ghys construction of the hyperbolic metric on the space of polygons with fixed directions of edges. The space of convex d -dimensional polyhedra with fixed directions of facet normals has a decomposition into type cones that correspond to different combinatorial types of polyhedra. This decomposition is a subfan of the secondary fan of a vector configuration and can be analyzed with the help of Gale diagrams. We construct a family of quadratic forms on each of the type cones using the theory of mixed volumes. The Alexandrov–Fenchel inequalities ensure that these forms have exactly one positive eigenvalue. This introduces a piecewise hyperbolic structure on the space of similarity classes of polyhedra with fixed directions of facet normals. We show that some of the dihedral angles on the boundary of the resulting cone-manifold are equal to π/2
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