Transactions of the American Mathematical Society, 1989
Let L be a line bundle on an abstract nonsingular curve C, let V c H°{C,L) be a linear system, an... more Let L be a line bundle on an abstract nonsingular curve C, let V c H°{C,L) be a linear system, and denote by C(d) the symmetric product of d copies of C . There exists a canonically defined C(d)-bundle map: where EL is a bundle of rank d obtained from L by a so-called symmetrization process. The various degenerary loci of a can be considered as subsecant schemes of CW . Our main result, Theorem 4.2, is given in §4, where we obtain a local matrix description of a valid (also) at points on the diagonal in C(rf', and thereby we can determine the completions of the local rings of the secant schemes at arbitrary points. In §5 we handle the special case of giving a local scheme structure to the zero set of a .
The São Paulo Journal of Mathematical Sciences, Jul 7, 2022
We consider q-matroids and their associated classical matroids derived from Gabidulin rank-metric... more We consider q-matroids and their associated classical matroids derived from Gabidulin rank-metric codes. We express the generalized rank weights of a Gabidulin rank-metric code in terms of Betti numbers of the dual classical matroid associated to the q-matroid corresponding to the code. In our main result, we show how these Betti numbers and their elongations determine the generalized weight polynomials for q-matroids, in particular, for the Gabidulin rank-metric codes. In addition, we demonstrate how the weight distribution and higher weight spectra of such codes can be determined directly from the associated q-matroids by using Möbius functions of its lattice of q-flats.
In this paper we study various aspects concerning almost ane codes, a class including, and strict... more In this paper we study various aspects concerning almost ane codes, a class including, and strictly larger than, that of linear codes. We use the combinatorial tool demi-matroids to show how one can dene relative length/dimension and dimension/length proles of ags(chains) of almost ane codes. In addition we show two specic results. One such result is how one can express the relative length/dimension proles (also called relative generalized Hamming weights) of a pair of codes in terms of intersection properties between the smallest of these codes and subcodes of the largest code. The other result tells how one can nd the extended weight polynomials, expressing the number of codewords of each possible weight, for each code in an innite hierarchy of extensions of a code over a given alphabet.
We introduce greedy weights of matroids, inspired by those for linear codes. We show that a Wei d... more We introduce greedy weights of matroids, inspired by those for linear codes. We show that a Wei duality holds for two of these types of greedy weights for matroids. Moreover we show that in the cases where the matroids involved are associated to linear codes, our definitions coincide with those for codes. Thus our Wei duality is a generalization of that for linear codes given by Schaathun. In the last part of the paper we show how some important chains of cycles of the matroids appearing, correspond to chains of component maps of minimal resolutions of the independence complex of the corresponding matroids. We also relate properties of these resolutions to chainedness and greedy weights of the matroids, and in many cases codes, that appear.
Local multiplicities of tangential trisecants to space curves
Springer eBooks, 1990
ABSTRACT Let C be a curve in P K 3 , where K is an algebraically closed field of characteristic z... more ABSTRACT Let C be a curve in P K 3 , where K is an algebraically closed field of characteristic zero. Assume that C is irreducible, reduced, and that C has no cusps. It is then known that C possesses finitely many tantential trisecants, and that the number of tangential trisecants is 2(d-2)(d-3)+2g(d-6) where d is the degree of C, and g is the geometric genus. The last statement is only correct if one counts tangential trisecants, including flexes, nodes, bitangents etc., with their proper multiplicities. We show how one can do this by studying an intersection product of the weak diagonal and a certain determinantal variety in the third symmetric product of the normalization of C. If C possesses cupsps, then there will be an excess component of intersection for last product. We use the set-up from [F] to show how one can find the contribution of cusps to the global number of tangential trisicants.
Secant lines of smooth projective curves; an infinitesimal study of the symmetric products
Contemporary mathematics, 1991
Local multiplicities of tangential trisecants to space curves
Lecture Notes in Mathematics, 1990
ABSTRACT Let C be a curve in P K 3 , where K is an algebraically closed field of characteristic z... more ABSTRACT Let C be a curve in P K 3 , where K is an algebraically closed field of characteristic zero. Assume that C is irreducible, reduced, and that C has no cusps. It is then known that C possesses finitely many tantential trisecants, and that the number of tangential trisecants is 2(d-2)(d-3)+2g(d-6) where d is the degree of C, and g is the geometric genus. The last statement is only correct if one counts tangential trisecants, including flexes, nodes, bitangents etc., with their proper multiplicities. We show how one can do this by studying an intersection product of the weak diagonal and a certain determinantal variety in the third symmetric product of the normalization of C. If C possesses cupsps, then there will be an excess component of intersection for last product. We use the set-up from [F] to show how one can find the contribution of cusps to the global number of tangential trisicants.
Transactions of the American Mathematical Society, 1986
Let C be a curve in P3 over an algebraically closed field of characteristic zero. We assume that ... more Let C be a curve in P3 over an algebraically closed field of characteristic zero. We assume that C is nonsingular and contains no plane component except possibly an irreducible conic. In [GP] one defines closed r-secant varieties to C, r £ N. These varieties are embedded in G, the Grassmannian of lines in P3. Denote by T the 3-secant variety (curve), and assume that the set of 4-secants is finite. Let T be the curve obtained by blowing up the ideal of 4-secants in T. The curve T is in general not in G. We study the local geometry of T at any point whose fibre of the blowingup map is reduced at the point. The multiplicity of T at such a point is determined in terms of the local geometry of C at certain chosen secant points. Furthermore we give a geometrical interpretation of the tangential directions of T at a singular point. We also give a criterion for whether all the tangential directions are distinct or not.
ABSTRACT We study algebraic geometric codes obtained from rational normal scrolls. We determine t... more ABSTRACT We study algebraic geometric codes obtained from rational normal scrolls. We determine the complete weight hierarchy and spectrum of these codes.
We study q-ary linear codes C obtained from Veronese surfaces over finite fields. We show how one... more We study q-ary linear codes C obtained from Veronese surfaces over finite fields. We show how one can find the higher weight spectra of these codes, or equivalently, the weight distribution of all extension codes of C over all field extensions of Fq. Our methods will be a study of the Stanley-Reisner rings of a series of matroids associated to each code C.
We define and study a class of Reed-Muller type errorcorrecting codes obtained from elementary sy... more We define and study a class of Reed-Muller type errorcorrecting codes obtained from elementary symmetric functions in finitely many variables. We determine the code parameters and higher weight spectra in the simplest cases.
We study q-ary linear codes C obtained from Veronese surfaces over finite fields. We show how one... more We study q-ary linear codes C obtained from Veronese surfaces over finite fields. We show how one can find the higher weight spectra of these codes, or equivalently, the weight distribution of all extension codes of C over all field extensions of Fq. Our methods will be a study of the Stanley-Reisner rings of a series of matroids associated to each code C.
Applicable Algebra in Engineering, Communication and Computing, 2013
To each linear code C over a finite field we associate the matroid M (C) of its parity check matr... more To each linear code C over a finite field we associate the matroid M (C) of its parity check matrix. For any matroid M one can define its generalized Hamming weights, and if a matroid is associated to such a parity check matrix, and thus of type M (C), these weights are the same as those of the code C. In our main result we show how the weights d1, • • • , d k of a matroid M are determined by the N-graded Betti numbers of the Stanley-Reisner ring of the simplicial complex whose faces are the independent sets of M , and derive some consequences. We also give examples which give negative results concerning other types of (global) Betti numbers, and using other examples we show that the generalized Hamming weights do not in general determine the N-graded Betti numbers of the Stanley-Reisner ring. The negative examples all come from matroids of type M (C).
We consider q-matroids and their associated classical matroids derived from Gabidulin rank-metric... more We consider q-matroids and their associated classical matroids derived from Gabidulin rank-metric codes. We express the generalized rank weights of a Gabidulin rank-metric code in terms of Betti numbers of the dual classical matroid associated to the q-matroid corresponding to the code. In our main result, we show how these Betti numbers and their elongations determine the generalized weight polynomials for q-matorids, in particular, for the Gabidulin rank-metric codes. In addition, we demonstrate how the weight distribution and higher weight spectra of such codes can be determined directly from the associated q-matroids by using Möbius functions of its lattice of q-flats.
We consider the notion of a (q, m)-polymatroid, due to Shiromoto, and the more general notion of ... more We consider the notion of a (q, m)-polymatroid, due to Shiromoto, and the more general notion of (q, m)-demi-polymatroid, and show how generalized weights can be defined for them. Further, we establish a duality for these weights analogous to Wei duality for generalized Hamming weights of linear codes. The corresponding results of Ravagnani for Delsarte rank metric codes, and Martínez-Peñas and Matsumoto for relative generalized rank weights are derived as a consequence.
We study subsets of Grassmann varieties G(l, m) over a field F , such that these subsets are unio... more We study subsets of Grassmann varieties G(l, m) over a field F , such that these subsets are unions of Schubert cycles, with respect to a fixed flag. We study the linear spans of, and in case of positive characteristic, the number of Fq-rational points on such unions. Moreover we study a geometric duality of such unions, and give a combinatorial interpretation of this duality. We discuss the maximum number of Fq-rational points for Schubert unions of a given spanning dimension, and we give some applications to coding theory. We define Schubert union codes, and study the parameters and support weights of these codes and of the well-known Grassmann codes.
This paper is devoted to giving a generalization from linear codes to the larger class of almost ... more This paper is devoted to giving a generalization from linear codes to the larger class of almost affine codes of two different results. One such result is how one can express the relative generalized Hamming weights of a pair of codes in terms of intersection properties between the smallest of these codes and subcodes of the largest code. The other result tells how one can find the extended weight polynomials, expressing the number of codewords of each possible weight, for each code in an infinite hierarchy of extensions of a code over a given alphabet. Our tools will be demi-matroids and matroids.
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