Jessica Sidman

Prolongations and Computational Algebra

We explore the geometric notion of prolongations in the setting of computational algebra, extending results of Landsberg and Manivel which relate prolongations to equations for secant varieties. We also develop methods for computing prolongations that are combinatorial in nature. As an application, we use prolongations to derive a new family of secant equations for the binary symmetric model in phylogenetics. Department of Mathematics and Statistics, Mount Holyoke College, South Hadley, MA 01075, U.S.A. e-mail: jsidman@mtholyoke.edu Department of Mathematics, North Carolina State University, Raleigh, NC, USA e-mail: smsulli2@ncsu.edu Received by the editors November 21, 2006; revised April 3, 2007. Sidman was partially supported by NSF grant DMS-0600471 and the Clare Boothe Luce Program. AMS subject classification: Primary: 13P10; secondary: 14M99. c ©Canadian Mathematical Society 2009. 930

SUBSPACE ARRANGEMENTS DEFINED BY PRODUCTS OF LINEAR FORMS

The vanishing ideal of an arrangement of linear subspaces in a vector space is considered, and the paper investigates when this ideal can be generated by products of linear forms. A combinatorial construction (blocker duality) is introduced which yields such generators in cases with a great deal of combinatorial structure, and examples are presented that inspired the work. A construction is given which produces all elements of this type in the vanishing ideal of the arrangement. This leads to an algorithm for deciding if the ideal is generated by products of linear forms. Generic arrangements of points in and lines in are also considered.

Generic initial ideals of points and curves

Let I be the defining ideal of a smooth complete intersection space curve C with defining equations of degrees a and b. We use the partial elimination ideals introduced by Mark Green to show that the lexicographic generic initial ideal of I has Castelnuovo-Mumford regularity 1+ab(a-1)(b-1)/2 with the exception of the case a=b=2, where the regularity is 4. Note that ab(a-1)(b-1)/2 is exactly the number of singular points of a general projection of C to the plane. Additionally, we show that for any term ordering @t, the generic initial ideal of a generic set of points in P^r is a @t-segment ideal.

Secant varieties of toric varieties

Abstract Let X P be a smooth projective toric variety of dimension n embedded in P r using all of the lattice points of the polytope P . We compute the dimension and degree of the secant variety Sec X P . We also give explicit formulas in dimensions 2 and 3 and obtain partial results for the projective varieties X A embedded using a set of lattice points A ⊂ P ∩ Z n containing the vertices of P and their nearest neighbors.

Combinatorics and the rigidity of CAD systems

We study the rigidity of body-and-cad frameworks which capture the majority of the geometric constraints used in 3D mechanical engineering CAD software. We present a combinatorial characterization of the generic minimal rigidity of a subset of body-and-cad frameworks in which we treat 20 of the 21 body-and-cad constraints, omitting only point-point coincidences. While the handful of classical combinatorial characterizations of rigidity focus on distance constraints between points, this is the first result simultaneously addressing coincidence, angular, and distance constraints. Our result is stated in terms of the partitioning of a graph into edge-disjoint spanning trees. This combinatorial approach provides the theoretical basis for the development of deterministic algorithms (that will not depend on numerical methods) for analyzing the rigidity of body-and-cad frameworks.

Syzygies of the secant variety of a curve

We show that the secant variety of a linearly normal smooth curve of degree at least 2g+3 is arithmetically Cohen-Macaulay, and we use this information to study the graded Betti numbers of the secant variety.

Castelnuovo–Mumford regularity by approximation

Abstract The Castelnuovo–Mumford regularity of a module gives a rough measure of its complexity. We bound the regularity of a module given a system of approximating modules whose regularities are known. Such approximations can arise naturally for modules constructed by inductive combinatorial means. We apply these methods to bound the regularity of ideals constructed as combinations of linear ideals and the module of derivations of a hyperplane arrangement as well as to give degree bounds for invariants of finite groups.

EQUATIONS DEFINING SECANT VARIETIES: GEOMETRY AND COMPUTATION

In the 1980’s, work of Green and Lazarsfeld (Invent. Math., 83, 1 (1985), 73–90; Compositio Math., 67, 3 (1988), 301–314), helped to uncover the beautiful interplay between the geometry of the embedding of a curve and the syzygies of its defining equations. Similar results hold for the first secant variety of a curve, and there is a natural conjectural picture extending to higher secant varieties as well. We present an introduction to the algebra and geometry used in (Sidman and Vermeire, Algebra Number Theory, 3, 4 (2009), 445–465) to study syzygies of secant varieties of curves with an emphasis on examples of explicit computations and elementary cases that illustrate the geometric principles at work.

Lectures on the Geometry of Syzygies

The theory of syzygies connects the qualitative study of algebraic varieties and commutative rings with the study of their defining equations. It started with Hilbert’s work on what we now call the Hilbert function and polynomial, and is important in our day in many new ways, from the high abstractions of derived equivalences to the explicit computations made possible by Gröbner bases. These lectures present some highlights of these interactions, with a focus on concrete invariants of syzygies that reflect basic invariants of algebraic sets.

Linear determinantal equations for all projective schemes

We prove that every projective embedding of a connected scheme determined by the complete linear series of a sufficiently ample line bun dle is defined by the 2 � 2 minors of a 1-generic matrix of linear forms. Extending the work of Eisenbud-Koh-Stillman for integral curves, we also provide effective descriptions fo r such determinantally presented ample line bundles on products of projective spaces, Gorenstein toric varieties, and smooth n-folds.