Peter Vermeire

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.

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.

Equations and syzygies of the first secant variety to a smooth curve

We show that if C is a linearly normal smooth curve embedded by a line bundle of degree at least 2g+3+p then the secant variety to the curve satisfies N_{3,p}.

Betti tables of reducible algebraic curves

We study the Betti tables of reducible algebraic curves with a focus on connected line arrangements and provide a general formula for computing the quadratic strand of the Betti table for line arrangements that satisfy certain hypotheses. We also give explicit formulas for the entries of the Betti tables for all curves of genus zero and one. Last, we give formulas for the graded Betti numbers for a class of curves of higher genus.

Classes of stable complex matrices defined via the theorems of Geršgorin and Lyapunov

We consider (and characterize) mainly classes of (positively) stable complex matrices defined via methods of Geršgorin and Lyapunov. Although the real matrices in most of these classes have already been studied, we sometimes improve upon (and even correct) what has been previously published. Many of the classes turn out quite naturally to be the products of common sets of matrices. A Venn diagram shows how the classes are related.

THE PARTY OF INVARIANTS ON SMOOTH THREE-FOLD HYPERSURFACES #

ABSTRACT Motivated by Hartshornes work on curves in ℙ3, we study the invariants of coherent sheaves on projective threefold hypersurfaces.