241 research outputs found
Semicontinuity for representations of one-dimensional Cohen-Macaulay rings
We show that the number of parameters for CM-modules of prescribed rank is
semi-continuous in families of CM rings of Krull dimension 1. This transfers a
result of Knoerrer from the commutative to the not necessarily commutative
case. For this purpose we introduce the notion of ``dense subrings'' which
seems rather technical but, nevertheless, useful. It enables the construction
of ``almost versal'' families of modules for a given algebra and the definition
of the ``number of parameters''. The semi--continuity implies, in particular,
that the set of so-called ``wild algebras'' in any family is a countable union
of closed subsets. A very exciting problem is whether it is actually closed,
hence whether the set of tame algebras is open. However, together with the
results of a former paper of the authors the semi-continuity implies that tame
is indeed an open property for curve singularities (commutative CM rings). An
analogous procedure leads to the semicontinuity of the number of parameters in
other cases, like representations of finite dimensional algebras or finite
dimensional bimodules.Comment: LaTeX2
On Symplectic Coverings of the Projective Plane
We prove that a resolution of singularities of any finite covering of the
projective plane branched along a Hurwitz curve and, maybe, along a
line "at infinity" can be embedded as a symplectic submanifold into some
projective algebraic manifold equipped with an integer K\"{a}hler symplectic
form (assuming that if has negative nodes, then the covering is
non-singular over them). For cyclic coverings we can realize this embeddings
into a rational algebraic 3--fold. Properties of the Alexander polynomial of
are investigated and applied to the calculation of the first Betti
number of a resolution of singularities of
-sheeted cyclic coverings of branched along
and, maybe, along a line "at infinity". We prove that is even
if is an irreducible Hurwitz curve but, in contrast to the algebraic
case, that it can take any non-negative value in the case when
consists of several irreducible components.Comment: 42 page
An Axiomatic Setup for Algorithmic Homological Algebra and an Alternative Approach to Localization
In this paper we develop an axiomatic setup for algorithmic homological
algebra of Abelian categories. This is done by exhibiting all existential
quantifiers entering the definition of an Abelian category, which for the sake
of computability need to be turned into constructive ones. We do this
explicitly for the often-studied example Abelian category of finitely presented
modules over a so-called computable ring , i.e., a ring with an explicit
algorithm to solve one-sided (in)homogeneous linear systems over . For a
finitely generated maximal ideal in a commutative ring we
show how solving (in)homogeneous linear systems over can be
reduced to solving associated systems over . Hence, the computability of
implies that of . As a corollary we obtain the computability
of the category of finitely presented -modules as an Abelian
category, without the need of a Mora-like algorithm. The reduction also yields,
as a by-product, a complexity estimation for the ideal membership problem over
local polynomial rings. Finally, in the case of localized polynomial rings we
demonstrate the computational advantage of our homologically motivated
alternative approach in comparison to an existing implementation of Mora's
algorithm.Comment: Fixed a typo in the proof of Lemma 4.3 spotted by Sebastian Posu
The Ernst equation and ergosurfaces
We show that analytic solutions \mcE of the Ernst equation with non-empty
zero-level-set of \Re \mcE lead to smooth ergosurfaces in space-time. In
fact, the space-time metric is smooth near a "Ernst ergosurface" if and
only if \mcE is smooth near and does not have zeros of infinite order
there.Comment: 23 pages, 4 figures; misprints correcte
A Purely Functional Computer Algebra System Embedded in Haskell
We demonstrate how methods in Functional Programming can be used to implement
a computer algebra system. As a proof-of-concept, we present the
computational-algebra package. It is a computer algebra system implemented as
an embedded domain-specific language in Haskell, a purely functional
programming language. Utilising methods in functional programming and prominent
features of Haskell, this library achieves safety, composability, and
correctness at the same time. To demonstrate the advantages of our approach, we
have implemented advanced Gr\"{o}bner basis algorithms, such as Faug\`{e}re's
and , in a composable way.Comment: 16 pages, Accepted to CASC 201
On the Milnor formula in arbitrary characteristic
The Milnor formula relates the Milnor number , the
double point number and the number of branches of a plane curve
singularity. It holds over the fields of characteristic zero. Melle and Wall
based on a result by Deligne proved the inequality in
arbitrary characteristic and showed that the equality
characterizes the singularities with no wild vanishing cycles. In this note we
give an account of results on the Milnor formula in characteristic . It
holds if the plane singularity is Newton non-degenerate (Boubakri et al. Rev.
Mat. Complut. (2010) 25) or if is greater than the intersection number of
the singularity with its generic polar (Nguyen H.D., Annales de l'Institut
Fourier, Tome 66 (5) (2016)). Then we improve our result on the Milnor number
of irreducible singularities (Bull. London Math. Soc. 48 (2016)). Our
considerations are based on the properties of polars of plane singularities in
characteristic .Comment: 18 page
Equianalytic and equisingular families of curves on surfaces
We consider flat families of reduced curves on a smooth surface S such that
each member C has the same number of singularities of fixed singularity types
and the corresponding (locally closed) subscheme H of the Hilbert scheme of S.
We are mainly concerned with analytic resp. topological singularity types and
give a sufficient condition for the smoothness of H (at C). Our results for
S=P^2 seem to be quite sharp for families of cuves of small degree d.Comment: LaTeX v 2.0
Invariants of Artinian Gorenstein Algebras and Isolated Hypersurface Singularities
We survey our recently proposed method for constructing biholomorphic
invariants of quasihomogeneous isolated hypersurface singularities and, more
generally, invariants of graded Artinian Gorenstein algebras. The method
utilizes certain polynomials associated to such algebras, called
nil-polynomials, and we compare them with two other classes of polynomials that
have also been used to produce invariants.Comment: 13 page
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