86 research outputs found
On surfaces in P^4 and 3-folds in P^5
This is a survey on the classification of smooth surfaces in P^4 and smooth
3-folds in P^5. We recall the corresponding results arising from adjunction
theory and explain how to construct examples via syzygies. We discuss some
examples in detail and list all families of smooth non-general type surfaces in
P^4 and 3-folds in P^5 known to us.Comment: 24 pages, AMS-TeX 2.
Local analysis of Grauert-Remmert-type normalization algorithms
Normalization is a fundamental ring-theoretic operation; geometrically it
resolves singularities in codimension one. Existing algorithmic methods for
computing the normalization rely on a common recipe: successively enlarge the
given ring in form an endomorphism ring of a certain (fractional) ideal until
the process becomes stationary. While Vasconcelos' method uses the dual
Jacobian ideal, Grauert-Remmert-type algorithms rely on so-called test ideals.
For algebraic varieties, one can apply such normalization algorithms
globally, locally, or formal analytically at all points of the variety. In this
paper, we relate the number of iterations for global Grauert-Remmert-type
normalization algorithms to that of its local descendants.
We complement our results by an explicit study of ADE singularities. This
includes the description of the normalization process in terms of value
semigroups of curves. It turns out that the intermediate steps produce only ADE
singularities and simple space curve singularities from the list of
Fruehbis-Krueger.Comment: 22 pages, 7 figure
Gr\"obner Bases over Algebraic Number Fields
Although Buchberger's algorithm, in theory, allows us to compute Gr\"obner
bases over any field, in practice, however, the computational efficiency
depends on the arithmetic of the ground field. Consider a field , a simple extension of , where is an
algebraic number, and let be the minimal polynomial of
. In this paper we present a new efficient method to compute Gr\"obner
bases in polynomial rings over the algebraic number field . Starting from
the ideas of Noro [Noro, 2006], we proceed by joining to the ideal to be
considered, adding as an extra variable. But instead of avoiding
superfluous S-pair reductions by inverting algebraic numbers, we achieve the
same goal by applying modular methods as in [Arnold, 2003; B\"ohm et al., 2015;
Idrees et al., 2011], that is, by inferring information in characteristic zero
from information in characteristic . For suitable primes , the
minimal polynomial is reducible over . This allows us to
apply modular methods once again, on a second level, with respect to the
factors of . The algorithm thus resembles a divide and conquer strategy and
is in particular easily parallelizable. At current state, the algorithm is
probabilistic in the sense that, as for other modular Gr\"obner basis
computations, an effective final verification test is only known for
homogeneous ideals or for local monomial orderings. The presented timings show
that for most examples, our algorithm, which has been implemented in SINGULAR,
outperforms other known methods by far.Comment: 16 pages, 1 figure, 1 tabl
Syzygies of Abelian and Bielliptic Surfaces in P^4
So far only six families of smooth irregular surfaces are known to exist in
P^4 (up to pullbacks by suitable finite covers of P^4). These are the elliptic
quintic scrolls, the minimal abelian and bielliptic surfaces (of degree 10),
two different families of non-minimal abelian surfaces of degree 15, and one
family of non-minimal bielliptic surfaces of degree 15.
The main purpose of the paper is to describe the structure of the
Hartshorne-Rao modules and the syzygies for each of these smooth irregular
surfaces in P^4, providing at the same time a unified construction method (via
syzygies) for these families of surfaces.Comment: 64 pages, author-supplied DVI file available at
http://oscar.math.brandeis.edu/~popescu/dvi/bielliptics2.dvi AmS-TeX v. 2.
Parallel algorithms for normalization
Given a reduced affine algebra A over a perfect field K, we present parallel
algorithms to compute the normalization \bar{A} of A. Our starting point is the
algorithm of Greuel, Laplagne, and Seelisch, which is an improvement of de
Jong's algorithm. First, we propose to stratify the singular locus Sing(A) in a
way which is compatible with normalization, apply a local version of the
normalization algorithm at each stratum, and find \bar{A} by putting the local
results together. Second, in the case where K = Q is the field of rationals, we
propose modular versions of the global and local-to-global algorithms. We have
implemented our algorithms in the computer algebra system SINGULAR and compare
their performance with that of the algorithm of Greuel, Laplagne, and Seelisch.
In the case where K = Q, we also discuss the use of modular computations of
Groebner bases, radicals, and primary decompositions. We point out that in most
examples, the new algorithms outperform the algorithm of Greuel, Laplagne, and
Seelisch by far, even if we do not run them in parallel.Comment: 19 page
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