56 research outputs found
Isohedra with dart-shaped faces
AbstractA polyhedron in E3 is said to be isohedral (or an isohedron) if its faces are equivalent under the action of its group of symmetries. We use Möbius nets of the three reflection groups of the five Platonic solids to construct isohedra whose faces are dart-shaped, and whose edges lie in planes of reflective symmetry of the polyhedron. This technique for constructing isohedra has only recently been used; it yields many new results in addition to those described in this paper. In the final section we also describe some other isohedra with dart-shaped faces
Elliptic curve configurations on Fano surfaces
The elliptic curves on a surface of general type constitute an obstruction
for the cotangent sheaf to be ample. In this paper, we give the classification
of the configurations of the elliptic curves on the Fano surface of a smooth
cubic threefold. That means that we give the number of such curves, their
intersections and a plane model. This classification is linked to the
classification of the automorphism groups of theses surfaces.Comment: 17 pages, accepted and shortened version, the rest will appear in
"Fano surfaces with 12 or 30 elliptic curves
Quantum affine Cartan matrices, Poincare series of binary polyhedral groups, and reflection representations
We first review some invariant theoretic results about the finite subgroups
of SU(2) in a quick algebraic way by using the McKay correspondence and quantum
affine Cartan matrices. By the way it turns out that some parameters
(a,b,h;p,q,r) that one usually associates with such a group and hence with a
simply-laced Coxeter-Dynkin diagram have a meaningful definition for the
non-simply-laced diagrams, too, and as a byproduct we extend Saito's formula
for the determinant of the Cartan matrix to all cases. Returning to invariant
theory we show that for each irreducible representation i of a binary
tetrahedral, octahedral, or icosahedral group one can find a homomorphism into
a finite complex reflection group whose defining reflection representation
restricts to i.Comment: 19 page
Reconstructing sparticle mass spectra using hadronic decays
Most sparticle decay cascades envisaged at the Large Hadron Collider (LHC) involve hadronic decays of intermediate particles. We use state-of-the art techniques based on the K⊥ jet algorithm to reconstruct the resulting hadronic final states for simulated LHC events in a number of benchmark supersymmetric scenarios. In particular, we show that a general method of selecting preferentially boosted massive particles such as W±, Z0 or Higgs bosons decaying to jets, using sub-jets found by the K⊥ algorithm, suppresses QCD backgrounds and thereby enhances the observability of signals that would otherwise be indistinct. Consequently, measurements of the supersymmetric mass spectrum at the per-cent level can be obtained from cascades including the hadronic decays of such massive intermediate bosons
On Proper Polynomial Maps of
Two proper polynomial maps are said to be \emph{equivalent} if there exist such that .
We investigate proper polynomial maps of arbitrary topological degree up to equivalence. Under the further assumption that the maps are Galois
coverings we also provide the complete description of equivalence classes. This
widely extends previous results obtained by Lamy in the case .Comment: 15 pages. Final version, to appear in Journal of Geometric Analysi
Lyashko-Looijenga morphisms and submaximal factorisations of a Coxeter element
When W is a finite reflection group, the noncrossing partition lattice NCP_W
of type W is a rich combinatorial object, extending the notion of noncrossing
partitions of an n-gon. A formula (for which the only known proofs are
case-by-case) expresses the number of multichains of a given length in NCP_W as
a generalised Fuss-Catalan number, depending on the invariant degrees of W. We
describe how to understand some specifications of this formula in a case-free
way, using an interpretation of the chains of NCP_W as fibers of a
Lyashko-Looijenga covering (LL), constructed from the geometry of the
discriminant hypersurface of W. We study algebraically the map LL, describing
the factorisations of its discriminant and its Jacobian. As byproducts, we
generalise a formula stated by K. Saito for real reflection groups, and we
deduce new enumeration formulas for certain factorisations of a Coxeter element
of W.Comment: 18 pages. Version 2 : corrected typos and improved presentation.
Version 3 : corrected typos, added illustrated example. To appear in Journal
of Algebraic Combinatoric
Zonotopes and four-dimensional superconformal field theories
The a-maximization technique proposed by Intriligator and Wecht allows us to
determine the exact R-charges and scaling dimensions of the chiral operators of
four-dimensional superconformal field theories. The problem of existence and
uniqueness of the solution, however, has not been addressed in general setting.
In this paper, it is shown that the a-function has always a unique critical
point which is also a global maximum for a large class of quiver gauge theories
specified by toric diagrams. Our proof is based on the observation that the
a-function is given by the volume of a three dimensional polytope called
"zonotope", and the uniqueness essentially follows from Brunn-Minkowski
inequality for the volume of convex bodies. We also show a universal upper
bound for the exact R-charges, and the monotonicity of a-function in the sense
that a-function decreases whenever the toric diagram shrinks. The relationship
between a-maximization and volume-minimization is also discussed.Comment: 29 pages, 15 figures, reference added, typos corrected, version
published in JHE
Moduli of Abelian varieties, Vinberg theta-groups, and free resolutions
We present a systematic approach to studying the geometric aspects of Vinberg
theta-representations. The main idea is to use the Borel-Weil construction for
representations of reductive groups as sections of homogeneous bundles on
homogeneous spaces, and then to study degeneracy loci of these vector bundles.
Our main technical tool is to use free resolutions as an "enhanced" version of
degeneracy loci formulas. We illustrate our approach on several examples and
show how they are connected to moduli spaces of Abelian varieties. To make the
article accessible to both algebraists and geometers, we also include
background material on free resolutions and representation theory.Comment: 41 pages, uses tabmac.sty, Dedicated to David Eisenbud on the
occasion of his 65th birthday; v2: fixed some typos and added reference
Liouville integrability of a class of integrable spin Calogero-Moser systems and exponents of simple Lie algebras
In previous work, we introduced a class of integrable spin Calogero-Moser
systems associated with the classical dynamical r-matrices with spectral
parameter, as classified by Etingof and Varchenko for simple Lie algebras. Here
the main purpose is to establish the Liouville integrability of these systems
by a uniform method
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