56 research outputs found
New constructions for covering designs
A {\em covering design}, or {\em covering}, is a family of
-subsets, called blocks, chosen from a -set, such that each -subset is
contained in at least one of the blocks. The number of blocks is the covering's
{\em size}, and the minimum size of such a covering is denoted by .
This paper gives three new methods for constructing good coverings: a greedy
algorithm similar to Conway and Sloane's algorithm for lexicographic
codes~\cite{lex}, and two methods that synthesize new coverings from
preexisting ones. Using these new methods, together with results in the
literature, we build tables of upper bounds on for , , and .
Depth of cohomology support loci for quasi-projective varieties via orbifold pencils
The present paper describes a relation between the quotient of the
fundamental group of a smooth quasi-projective variety by its second commutator
and the existence of maps to orbifold curves. It extends previously studied
cases when the target was a smooth curve. In the case when the quasi-projective
variety is a complement to a plane algebraic curve this provides new relations
between the fundamental group, the equation of the curve, and the existence of
polynomial solutions to certain equations generalizing Pell's equation. These
relations are formulated in terms of the depth which is an invariant of the
characters of the fundamental group discussed in detail here.Comment: 22 page
Stacks of cyclic covers of projective spaces
We define stacks of uniform cyclic covers of Brauer-Severi schemes, proving
that they can be realized as quotient stacks of open subsets of
representations, and compute the Picard group for the open substacks
parametrizing smooth uniform cyclic covers. Moreover, we give an analogous
description for stacks parametrizing triple cyclic covers of Brauer-Severi
schemes of rank 1, which are not necessarily uniform, and give a presentation
of the Picard group for substacks corresponding to smooth triple cyclic covers.Comment: 23 pages; some minor changes; to appear in Compositio Mathematic
Smoothness of the truncated display functor
We show that to every p-divisible group over a p-adic ring one can associate
a display by crystalline Dieudonne theory. For an appropriate notion of
truncated displays, this induces a functor from truncated Barsotti-Tate groups
to truncated displays, which is a smooth morphism of smooth algebraic stacks.
As an application we obtain a new proof of the equivalence between
infinitesimal p-divisible groups and nilpotent displays over p-adic rings, and
a new proof of the equivalence due to Berthelot and Gabber between commutative
finite flat group schemes of p-power order and Dieudonne modules over perfect
rings.Comment: 38 page
Dimers, Tilings and Trees
Generalizing results of Temperley, Brooks, Smith, Stone and Tutte and others
we describe a natural equivalence between three planar objects: weighted
bipartite planar graphs; planar Markov chains; and tilings with convex
polygons. This equivalence provides a measure-preserving bijection between
dimer coverings of a weighted bipartite planar graph and spanning trees on the
corresponding Markov chain. The tilings correspond to harmonic functions on the
Markov chain and to ``discrete analytic functions'' on the bipartite graph.
The equivalence is extended to infinite periodic graphs, and we classify the
resulting ``almost periodic'' tilings and harmonic functions.Comment: 23 pages, 5 figure
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