230 research outputs found
A characteristic free criterion of birationality
One develops {\em ab initio} the theory of rational/birational maps over
reduced, but not necessarily irreducible, projective varieties in arbitrary
characteristic. A numerical invariant of a rational map is introduced, called
the Jacobian dual rank. It is proved that a rational map in this general setup
is birational if and only if the Jacobian dual rank attains its maximal
possible value. Even in the "classical" case where the source variety is
irreducible there is some gain for this invariant over the degree of the map as
it is, on one hand, intrinsically related to natural constructions in
commutative algebra and, on the other hand, is effectively straightforwardly
computable. Applications are given to results so far only known in
characteristic zero. In particular, the surprising result of Dolgachev
concerning the degree of a plane polar Cremona map is given an alternative
conceptual angle.Comment: 24 page
Algebraic methods for parameterized codes and invariants of vanishing ideals over finite fields
Let K be a finite field with q elements and let X be a subset of a projective
space P^{s-1}, over the field K, which is parameterized by Laurent monomials.
Let I(X) be the vanishing ideal of X. Some of the main contributions of this
paper are in determining the structure of I(X) and some of their invariants. It
is shown that I(X) is a lattice ideal. We introduce the notion of a
parameterized code arising from X and present algebraic methods to compute and
study its dimension, length and minimum distance. For a parameterized code
arising from a connected graph we are able to compute its length and to make
our results more precise. If the graph is non-bipartite, we show an upper bound
for the minimum distance. We also study the underlying geometric structure of
X.Comment: Finite Fields Appl., to appea
Monomial transformations of the projective space
We prove that, over any field, the dimension of the indeterminacy locus of a
rational transformation of which is defined by monomials of the same
degree with no common factors is at least , provided that the
degree of as a map is not divisible by . This implies upper bounds on
the multidegree of
Plane Cremona maps: saturation and regularity of the base ideal
One studies plane Cremona maps by focusing on the ideal theoretic and
homological properties of its homogeneous base ideal ("indeterminacy locus").
The {\em leitmotiv} driving a good deal of the work is the relation between the
base ideal and its saturation. As a preliminary one deals with the homological
features of arbitrary codimension 2 homogeneous ideals in a polynomial ring in
three variables over a field which are generated by three forms of the same
degree. The results become sharp when the saturation is not generated in low
degrees, a condition to be given a precise meaning. An implicit goal,
illustrated in low degrees, is a homological classification of plane Cremona
maps according to the respective homaloidal types. An additional piece of this
work relates the base ideal of a rational map to a few additional homogeneous
"companion" ideals, such as the integral closure, the -fat
ideal and a seemingly novel ideal defined in terms of valuations.Comment: New version only 36 pages, one typo correcte
Symbolic powers of monomial ideals and Cohen-Macaulay vertex-weighted digraphs
In this paper we study irreducible representations and symbolic Rees algebras
of monomial ideals. Then we examine edge ideals associated to vertex-weighted
oriented graphs. These are digraphs having no oriented cycles of length two
with weights on the vertices. For a monomial ideal with no embedded primes we
classify the normality of its symbolic Rees algebra in terms of its primary
components. If the primary components of a monomial ideal are normal, we
present a simple procedure to compute its symbolic Rees algebra using Hilbert
bases, and give necessary and sufficient conditions for the equality between
its ordinary and symbolic powers. We give an effective characterization of the
Cohen--Macaulay vertex-weighted oriented forests. For edge ideals of transitive
weighted oriented graphs we show that Alexander duality holds. It is shown that
edge ideals of weighted acyclic tournaments are Cohen--Macaulay and satisfy
Alexander dualityComment: Special volume dedicated to Professor Antonio Campillo, Springer, to
appea
Extending Landsat 8: Retrieval of an Orange contra-Band for Inland Water Quality Applications
The Operational Land Imager (OLI) onboard Landsat 8 has found successful application in inland and coastal water remote sensing. Its radiometric specification and high spatial resolution allows quantification of water-leaving radiance while resolving small water bodies. However, its limited multispectral band set restricts the range of water quality parameters that can be retrieved. Identification of cyanobacteria biomass has been demonstrated for sensors with a band centered near 620 nm, the absorption peak of the diagnostic pigment phycocyanin. While OLI lacks such a band in the orange region, superposition of the available multispectral and panchromatic bands suggests that it can be calculated by a scaled difference. A set of 428 in situ spectra acquired in diverse lakes in Belgium and The Netherlands was used to develop and test an orange contra-band retrieval algorithm, achieving a mean absolute percentage error of 5.39 % and a bias of −0.88 % in the presence of sensor noise. Atmospheric compensation error propagated to the orange contra-band was observed to maintain about the same magnitude (13 % higher) observed for the red band and thus results in minimal additional effects for possible base line subtraction or band ratio algorithms for phycocyanin estimation. Generality of the algorithm for different reflectance shapes was tested against a set of published average coastal and inland Optical Water Types, showing robust retrieval for all but relatively clear water types (Secchi disk depth > 6 m and chlorophyll a < 1.6 mg m−3). The algorithm was further validated with 79 matchups against the Ocean and Land Colour Imager (OLCI) orange band for 10 globally distributed lakes. The retrieved band is shown to convey information independent from the adjacent bands under variable phycocyanin concentrations. An example application using Landsat 8 imagery is provided for a knowncyanobacterialbloominLakeErie,US.ThemethodisdistributedintheACOLITEatmospheric correction code. The contra-band approach is generic and can be applied to other sensors with overlapping bands. Recommendations are also provided for development of future sensors with broad spectral bands with the objective to maximize the accuracy of possible spectral enhancement
Combinatorics of a certain ideal in the Segre coordinate ring
We focus on a 'fat' model of an ideal in the class of the canonical ideal of the Segre coordinate ring, looking at its Rees algebra and related arithmetical questions.124113285329
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