Points In Affine Space Research Articles

Frieze varieties: A characterization of the finite-tame-wild trichotomy for acyclic quivers

We introduce a new class of algebraic varieties which we call frieze varieties. Each frieze variety is determined by an acyclic quiver. The frieze variety is defined in an elementary recursive way by constructing a set of points in affine space. From a more conceptual viewpoint, the coordinates of these points are specializations of cluster variables in the cluster algebra associated to the quiver.We give a new characterization of the finite–tame–wild trichotomy for acyclic quivers in terms of their frieze varieties. We show that an acyclic quiver is representation finite, tame, or wild, respectively, if and only if the dimension of its frieze variety is 0,1, or ≥2, respectively.

Wonderful compactifications of the moduli space of points in affine and projective space

We introduce and study smooth compactifications of the moduli space of n labeled points with weights in projective space, which have normal crossings boundary and are defined as GIT quotients of the weighted Fulton–MacPherson compactification. We show that the GIT quotient of a wonderful compactification is also a wonderful compactification under certain hypotheses. We also study a weighted version of the configuration spaces parametrizing n points in affine space up to translation and homothety. In dimension one, the above compactifications are isomorphic to Hassett’s moduli space of rational weighted stable curves.

Evaluation codes and weierstrass semigroups

We report on a construction of modified evaluation codes by Dias and Neves [1]. Evaluation codes are a type of Reed-Muller code, where the code is the image of a space of polynomial functions under the evaluation map on a given set of points in affine space, cf. [2]. In their work, Dias and Neves assign weights to the coordinates, namely a finite sequence of natural numbers. The goal is to compute the parameters of the code, especially aiming at the Minimum-DistanceSeparable property. Even with the simplest choice of points on which to evaluate, however, the problem is difficult, as it is related to the Frobenius problem [3]. In collaboration with Neves, we propose to establish a connection between the spaces of functions and sections of line bundles on algebraic curves when the chosen sequence generates the Weierstrass semigroup of a point on the curve. The prospective application is to compute the parameters of the code in some cases by using Riemann-Roch arguments.

The vanishing ideal of a finite set of closed points in affine space

Given a finite set of closed rational points of affine space over a field, we give a Gröbner basis for the lexicographic ordering of the ideal of polynomials which vanish at all given points. Our method is an alternative to the Buchberger–Möller algorithm, but in contrast to that, we determine the set of leading terms of the ideal without solving any linear equation but by induction over the dimension of affine space. The elements of the Gröbner basis are also computed by induction over the dimension, using one-dimensional interpolation of coefficients of certain polynomials.

Computing zero-dimensional schemes

This paper is a natural continuation of Abbott et al. [Abbott, J., Bigatti, A., Kreuzer, M., Robbiano, L., 2000. Computing ideals of points. J. Symbolic Comput. 30, 341–356] further generalizing the Buchberger–Möller algorithm to zero-dimensional schemes in both affine and projective spaces. We also introduce a new, general way of viewing the problems which can be solved by the algorithm: an approach which looks to be readily applicable in several areas. Implementation issues are also addressed, especially for computations over Q where a trace-lifting paradigm is employed. We give a complexity analysis of the new algorithm for fat points in affine space over Q . Tables of timings show the new algorithm to be efficient in practice.