79 research outputs found
On the homology of the space of knots
Consider the space of `long knots' in R^n, K_{n,1}. This is the space of
knots as studied by V. Vassiliev. Based on previous work of the authors, it
follows that the rational homology of K_{3,1} is free Gerstenhaber-Poisson
algebra. A partial description of a basis is given here. In addition, the mod-p
homology of this space is a `free, restricted Gerstenhaber-Poisson algebra'.
Recursive application of this theorem allows us to deduce that there is
p-torsion of all orders in the integral homology of K_{3,1}.
This leads to some natural questions about the homotopy type of the space of
long knots in R^n for n>3, as well as consequences for the space of smooth
embeddings of S^1 in S^3.Comment: 36 pages, 6 figures. v3: small revisions before publicatio
From Galois to Hopf Galois: theory and practice
Hopf Galois theory expands the classical Galois theory by considering the
Galois property in terms of the action of the group algebra k[G] on K/k and
then replacing it by the action of a Hopf algebra. We review the case of
separable extensions where the Hopf Galois property admits a group-theoretical
formulation suitable for counting and classifying, and also to perform explicit
computations and explicit descriptions of all the ingredients involved in a
Hopf Galois structure. At the end we give just a glimpse of how this theory is
used in the context of Galois module theory for wildly ramified extensions
Nilpotent and abelian Hopf-Galois structures on field extensions
This is the author accepted manuscript. The final version is available from Elsevier via the DOI in this record.Let L/K be a finite Galois extension of fields with group Γ . When Γ is nilpotent, we show that the problem of enumerating all
nilpotent Hopf–Galois structures on L/K can be reduced to the corresponding problem for the Sylow subgroups of Γ . We use this to enumerate all nilpotent (resp. abelian) Hopf–Galois structures on a cyclic extension of arbitrary finite degree. When Γ is abelian, we give conditions under which every abelian Hopf–Galois structure on L/K has type Γ . We also give a criterion on n such that every Hopf–Galois structure on a cyclic extension of degree n has cyclic type
Mapping 6D N = 1 supergravities to F-theory
We develop a systematic framework for realizing general anomaly-free chiral
6D supergravity theories in F-theory. We focus on 6D (1, 0) models with one
tensor multiplet whose gauge group is a product of simple factors (modulo a
finite abelian group) with matter in arbitrary representations. Such theories
can be decomposed into blocks associated with the simple factors in the gauge
group; each block depends only on the group factor and the matter charged under
it. All 6D chiral supergravity models can be constructed by gluing such blocks
together in accordance with constraints from anomalies. Associating a geometric
structure to each block gives a dictionary for translating a supergravity model
into a set of topological data for an F-theory construction. We construct the
dictionary of F-theory divisors explicitly for some simple gauge group factors
and associated matter representations. Using these building blocks we analyze a
variety of models. We identify some 6D supergravity models which do not map to
integral F-theory divisors, possibly indicating quantum inconsistency of these
6D theories.Comment: 37 pages, no figures; v2: references added, minor typos corrected;
v3: minor corrections to DOF counting in section
Double covers and extensions
In this paper we consider double covers of the projective space in relation
with the problem of extensions of varieties, specifically of extensions of
canonical curves to surfaces and Fano 3-folds. In particular we consider
surfaces which are double covers of the plane branched over a general
sextic: we prove that the general curve in the linear system pull back of plane
curves of degree lies on a unique surface. If the
general such curve is instead extendable to a higher dimensional variety. In
the cases , this gives the existence of singular index Fano
varieties of dimensions 8, 5, 3, and genera 17, 26, 37 respectively. For we recover the Fano variety , one of only two Fano
threefolds with canonical Gorenstein singularities with the maximal genus 37,
found by Prokhorov. We show that the latter variety is no further extendable.
For and these Fano varieties have been identified by Totaro. We also
study the extensions of smooth degree 2 sections of surfaces of genus 3.
In all these cases, we compute the co-rank of the Gauss--Wahl maps of the
curves under consideration. Finally we observe that linear systems on double
covers of the projective plane provide superabundant logarithmic Severi
varieties.Comment: v2: added the Fano's found by Totaro; some references adde
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