Anthony V. Geramita

Ranks of tensors, secant varieties of Segre varieties and fat points

Abstract A classical unsolved problem of projective geometry is that of finding the dimensions of all the (higher) secant varieties of the Segre embeddings of an arbitrary product of projective spaces. An important subsidiary problem is that of finding the smallest integer t for which the secant variety of projective t -spaces fills the ambient projective space. In this paper we give a new approach to these problems. The crux of our method is the translation of a well-known lemma of Terracini into a question concerning the Hilbert function of “fat points” in a multiprojective space. Our approach gives much new information on the classical problem even in the case of three factors (a case also studied in the area of Algebraic Complexity Theory).

Monomial ideals and points in projective space

Let A=k[X0,…Xn]/I be the homogeneous co-ordinate ring of s points in generic position in Pn. The first and third authors have formulated natural conjectures for the number of generators of I, and for the Cohen-Macaulay type of A. In this paper we give a simple new proof of the conjectures for n = 2 (all s), and prove that the conjectures hold for ‘most’ s if n≥3.

The Hilbert function of a level algebra

Part 1. Nonexistence and Existence: Introduction Numerical conditions Homological methods Some refinements Constructing Artinian level algebras Constructing level sets of points Expected behavior Part 2. Appendix: A Classification of Codimension Three Level Algebras of Low Socle Degree: Appendix A. Introduction and notation Appendix B. Socle degree

Cayley-Bacharach schemes and their canonical modules

6

Geometric consequences of extremal behavior in a theorem of Macaulay

and Type

On the Cohen-Macaulay type of s-lines in An + 1

2

Secant varieties of Grassmann varieties

Appendix C. Socle degree

Level Algebras, Lex Segments, and Minimal Hilbert Functions

5

Reduced Gorenstein codimension three subschemes of projective space

Appendix D. Socle degree

Classifying Hilbert functions of fat point subschemes in ℙ2

4