27 research outputs found
On the total order of reducibility of a pencil of algebraic plane curves
In this paper, the problem of bounding the number of reducible curves in a
pencil of algebraic plane curves is addressed. Unlike most of the previous
related works, each reducible curve of the pencil is here counted with its
appropriate multiplicity. It is proved that this number of reducible curves,
counted with multiplicity, is bounded by d^2-1 where d is the degree of the
pencil. Then, a sharper bound is given by taking into account the Newton's
polygon of the pencil
The Cassels-Tate pairing on polarized abelian varieties
Let (A,\lambda) be a principally polarized abelian variety defined over a
global field k, and let \Sha(A) be its Shafarevich-Tate group. Let \Sha(A)_\nd
denote the quotient of \Sha(A) by its maximal divisible subgroup. Cassels and
Tate constructed a nondegenerate pairing \Sha(A)_\nd \times \Sha(A)_\nd
\rightarrow \Q/\Z. If A is an elliptic curve, then by a result of Cassels the
pairing is alternating. But in general it is only antisymmetric.
Using some new but equivalent definitions of the pairing, we derive general
criteria deciding whether it is alternating and whether there exists some
alternating nondegenerate pairing on \Sha(A)_\nd. These criteria are expressed
in terms of an element c \in \Sha(A)_\nd that is canonically associated to the
polarization \lambda. In the case that A is the Jacobian of some curve, a
down-to-earth version of the result allows us to determine effectively whether
\#\Sha(A) (if finite) is a square or twice a square. We then apply this to
prove that a positive proportion (in some precise sense) of all hyperelliptic
curves of even genus g \ge 2 over \Q have a Jacobian with nonsquare \#\Sha (if
finite). For example, it appears that this density is about 13% for curves of
genus 2. The proof makes use of a general result relating global and local
densities; this result can be applied in other situations.Comment: 41 pages, published versio
Progress in Commutative Algebra 2
This is the second of two volumes of a state-of-the-art survey article collection which originates from three commutative algebra sessions at the 2009 Fall Southeastern American Mathematical Society Meeting at Florida Atlantic University. The articles reach into diverse areas of commutative algebra and build a bridge between Noetherian and non-Noetherian commutative algebra. These volumes present current trends in two of the most active areas of commutative algebra: non-noetherian rings (factorization, ideal theory, integrality), and noetherian rings (the local theory, graded situation, and interactions with combinatorics and geometry). This volume contains surveys on aspects of closure operations, finiteness conditions and factorization. Closure operations on ideals and modules are a bridge between noetherian and nonnoetherian commutative algebra. It contains a nice guide to closure operations by Epstein, but also contains an article on test ideals by Schwede and Tucker and more
Iterative and Iterative-Noniterative Integral Solutions in 3-Loop Massive QCD Calculations
Various of the single scale quantities in massless and massive QCD up to
3-loop order can be expressed by iterative integrals over certain classes of
alphabets, from the harmonic polylogarithms to root-valued alphabets. Examples
are the anomalous dimensions to 3-loop order, the massless Wilson coefficients
and also different massive operator matrix elements. Starting at 3-loop order,
however, also other letters appear in the case of massive operator matrix
elements, the so called iterative non-iterative integrals, which are related to
solutions based on complete elliptic integrals or any other special function
with an integral representation that is definite but not a Volterra-type
integral. After outlining the formalism leading to iterative non-iterative
integrals,we present examples for both of these cases with the 3-loop anomalous
dimension and the structure of the principle solution in
the iterative non-interative case of the 3-loop QCD corrections to the
-parameter.Comment: 13 pages LATEX, 2 Figure
Equivariant Cox ring
We define the equivariant Cox ring of a normal variety with algebraic group
action. We study algebraic and geometric aspects of this object and show how it
is related to the usual Cox ring. Then, we specialize to the case of normal
rational varieties of complexity one under the action of a connected reductive
group G. We show that the equivariant Cox ring is finitely generated in this
case. Under a mild additional condition, we give a presentation by generators
and relations of its subalgebra of U-invariants, where U is the unipotent part
of a Borel subgroup of G. The ordinary Cox ring also has a canonical structure
of U-algebra, and we prove that the subalgebra of U-invariants is a finitely
generated Cox ring of a variety of complexity one under the action of a torus.
Using an earlier work from Hausen and Herppich, we obtain that this latter
algebra is a complete intersection. Iteration of Cox rings has been introduced
by Arzhantsev, Braun, Hausen and Wrobel in [1]. For log terminal quasicones
with a torus action of complexity one, they proved that the iteration sequence
is finite with a finitely generated factorial master Cox ring. We prove that
the iteration sequence is finite for equivariant and ordinary Cox rings of
normal rational G-varieties of complexity one satisfying a mild additional
condition (e.g. complete varieties or almost homogeneous varieties with only
constant invertible functions). In the almost homogeneous case, we prove that
the equivariant and ordinary master Cox rings are finitely generated and
factorial
Algebraic supergroups of Cartan type
I present a construction of connected affine algebraic supergroups G_V
associated with simple Lie superalgebras g of Cartan type and with g-modules V.
Conversely, I prove that every connected affine algebraic supergroup whose
tangent Lie superalgebra is of Cartan type is necessarily isomorphic to one of
the supergroups G_V that I introduced. In particular, the supergroup
constructed in this way associated with g := W(n) and its standard
representation is described somewhat more in detail.
In addition, *** an "Erratum" is added here *** after the main text to fix a
mistake which was kindly pointed out to the author by prof. Masuoka after the
paper was published: this "Erratum" is accepted for publication in "Forum
Mathematicum", it appears here in its final form (but prior to proofreading).
In it, I also explain more in detail the *Existence Theorem* for algebraic
supergroups of Cartan type which comes out of the main result in the original
paper.Comment: Main file: La-TeX file, 47 pages, already published (see below).
Erratum: La-TeX file, 6 pages, to appear (see below). For the main file, the
original publication is available at www.degruyter.com (cf. the journal
reference here below
Double Elliptic Dynamical Systems From Generalized Mukai - Sklyanin Algebras
We consider the double-elliptic generalisation of dynamical systems of
Calogero-Toda-Ruijsenaars type using finite-dimensional Mukai-Sklyanin
algebras. The two-body system, which involves an elliptic dependence both on
coordinates and momenta, is investigated in detail and the relation with Nambu
dynamics is mentioned. We identify the 2D complex manifold associated with the
double elliptic system as an elliptically fibered rational ("1/2K3 ") surface.
Some generalisations are suggested which provide the ground for a description
of the N-body systems. Possible applications to SUSY gauge theories with
adjoint matter in with two compact dimensions are discussed.Comment: 31 pages, Late