368 research outputs found
Linear Toric Fibrations
These notes are based on three lectures given at the 2013 CIME/CIRM summer
school. The purpose of this series of lectures is to introduce the notion of a
toric fibration and to give its geometrical and combinatorial
characterizations. Polarized toric varieties which are birationally equivalent
to projective toric bundles are associated to a class of polytopes called
Cayley polytopes. Their geometry and combinatorics have a fruitful interplay
leading to fundamental insight in both directions. These notes will illustrate
geometrical phenomena, in algebraic geometry and neighboring fields, which are
characterized by a Cayley structure. Examples are projective duality of toric
varieties and polyhedral adjunction theory
Self-dual projective toric varieties
Let T be a torus over an algebraically closed field k of characteristic 0,
and consider a projective T-module P(V). We determine when a projective toric
subvariety X of P(V) is self-dual, in terms of the configuration of weights of
V.Comment: 26 pages, 1 figure. Minor change
Describing codimension two defects
Codimension two defects of the six dimensional theory
have played an important role in the understanding
of dualities for certain SCFTs in four dimensions. These
defects are typically understood by their behaviour under various dimensional
reduction schemes. In their various guises, the defects admit partial
descriptions in terms of singularities of Hitchin systems, Nahm boundary
conditions or Toda operators. Here, a uniform dictionary between these
descriptions is given for a large class of such defects in
.Comment: 74pp, lots of tables detailing order reversing duality; (v2)
Acknowledgement added. Notation simplified, refs added, minor fixes ; (v3)
Minor changes, version accepted in JHEP. I thank the referee for helpful
comments towards improving presentatio
On linear series with negative Brill-Noether number
Brill-Noether theory studies the existence and deformations of curves in
projective spaces; its basic object of study is , the
moduli space of smooth genus curves with a choice of degree line bundle
having at least independent global sections. The Brill-Noether theorem
asserts that the map is
surjective with general fiber dimension given by the number , under the hypothesis that . One may
naturally conjecture that for , this map is generically finite onto a
subvariety of codimension in . This conjecture fails in
general, but seemingly only when is large compared to . This paper
proves that this conjecture does hold for at least one irreducible component of
, under the hypothesis that . We conjecture that this result should hold for all for some constant , and we give a purely combinatorial conjecture that
would imply this stronger result.Comment: 16 page
Immersed surfaces in the modular orbifold
A hyperbolic conjugacy class in the modular group PSL(2,Z) corresponds to a
closed geodesic in the modular orbifold. Some of these geodesics virtually
bound immersed surfaces, and some do not; the distinction is related to the
polyhedral structure in the unit ball of the stable commutator length norm. We
prove the following stability theorem: for every hyperbolic element of the
modular group, the product of this element with a sufficiently large power of a
parabolic element is represented by a geodesic that virtually bounds an
immersed surface.Comment: 13 pages, 8 figures; version 2 contains minor correction
Koszul algebras and regularity
This is a survey paper on commutative Koszul algebras and Castelnuovo-Mumford
regularity. We describe several techniques to establish the Koszulness of
algebras. We discuss variants of the Koszul property such as strongly Koszul,
absolutely Koszul and universally Koszul. We present several open problems
related with these notions and their local variants
On the cycle class map for zero-cycles over local fields
We study the Chow group of zero-cycles of smooth projective varieties over
local and strictly local fields. We prove in particular the injectivity of the
cycle class map to integral l-adic cohomology for a large class of surfaces
with positive geometric genus, over local fields of residue characteristic
different from l. The same statement holds for semistable K3 surfaces defined
over C((t)), but does not hold in general for surfaces over strictly local
fields.Comment: 37 pages (with an appendix by Spencer Bloch); bibliography updated,
final versio
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