This note introduces the Macaulay2 package SpaceCurves.m2 with illustration. The 1.0 version of the package, provided in the accompanying online supplement, is devoted to the generation of three types of curves in ސ 3 : smooth curves, ACM curves and curves that are minimal in the even liaison class.1. INTRODUCTION. The SpaceCurves project was initiated by R. Hartshorne, F. Schreyer and M. Stillman during the 2017 Macaulay2 workshop at UC Berkeley. Since the workshop, Zhang has improved old code and developed the package into the present version. The goal of the SpaceCurves package is to generate three types of curves in ސ 3 ; smooth curves, ACM curves, and minimal curves in a given even liaison class.In Section 2 we illustrate how to produce smooth curves exhausting all possibilities of (degree, genus) pairs. The philosophy is the following: first we construct three types of surfaces; the smooth quadric surface, smooth cubic surfaces and rational quartic surfaces with a double line. Next, we construct divisors on these surfaces. Finally, we generate a random curve in a given divisor class.In Section 3 we illustrate the stratification of the Hilbert scheme of ACM curves in ސ 3 using Betti tables. We illustrate how to produce ACM curves exhausting all possibilities of Betti tables. First, we list all the possible Hilbert functions of ACM curves of a given degree, then we produce all Betti tables of ACM curves with a given Hilbert function. Finally, we generate a matrix of random forms with degrees specified by the Hilbert-Burch degree matrix and take the ideal of maximal minors.In Section 4 we explain the construction of a curve that is minimal in its even liaison class from a given finite length module. The even liaison class can be specified by either the ideal of a curve in the even liaison class, or by a finite length module called the Hartshorne-Rao module. The implementation of the minimal curve algorithm follows the paper by Guarrera et al. [1997].To make our computations exact, and to avoid coefficient explosion, we work over large finite fields /ޚ pޚ in Macaulay2. Over a small prime field, such as ޚ2/ޚ
We study bundles on projective spaces that have vanishing lower cohomologies using their short minimal free resolutions. We classify the Betti numbers of these bundles and show that there are only finitely many possibilities with a given first Chern class and bounded regularity. Accordingly, we classify the Hilbert functions of these bundles and provide an efficient way to represent and to generate them. Finally, we show that the Betti numbers of bundles with a fixed Hilbert function form a graded lattice. We then describe the stratification of the space of isomorphism classes of bundles with a fixed Hilbert function by Betti numbers.
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