Alessandro Gimigliano

Computing symmetric rank for symmetric tensors

We consider the problem of determining the symmetric tensor rank for symmetric tensors with an algebraic geometry approach. We give algorithms for computing the symmetric rank for 2x...x2 tensors and for tensors of small border rank. From a geometric point of view, we describe the symmetric rank strata for some secant varieties of Veronese varieties.

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).

Secant varieties of Grassmann varieties

We consider the dimensions of the higher secant varieties of the Grassmann varieties. We give new instances where these secant varieties have the expected dimension and also a new example where a higher secant variety does not.

Regularity of linear systems of plane curves

The subject of this paper is the study of linear systems of plane curves which are given by assigning a set of fixed multiple points. This study not only has an interest per se, but is connected to other subjects (for example, results on rational surfaces and curves on them can be found in [Ro], [A], and [Gil). A linear system of plane curves of (projective) dimension r is a subvector space I, s

On the secant varieties to the tangential varieties of a Veronesean

We study the dimensions of the higher secant varieties to the tangent varieties of Veronese varieties. Our approach, generalizing that of Terracini, concerns 0-dimensional schemes which are the union of second infinitesimal neighbourhoods of generic points, each intersected with a generic double line. We find the deficient secant line varieties for all the Veroneseans and all the deficient higher secant varieties for the quadratic Veroneseans. We conjecture that these are the only deficient secant varieties in this family and prove this up to secant projective 4-spaces.

Betti numbers for fat point ideals in the plane: a geometric approach

We consider the open problem of determining the graded Betti numbers for fat point subschemes Z supported at general points of P 2 . We relate this problem to the open geometric problem of determining the splitting type of the pullback of P2 to the normalization of certain rational plane curves. We give a conjecture for the graded Betti numbers which would determine them in all degrees but one for every fat point subscheme supported at general points of P 2 . We also prove our Betti number conjecture in a broad range of cases. An appendix discusses many more cases in which our conjecture has been verified computationally and provides a new and more efficient computational approach for computing graded Betti numbers in certain degrees. It also demonstrates how to derive explicit conjectural values for the Betti numbers and how to compute splitting types.

Scrollar invariants and resolutions of certaind-gonal curves

In this note we compute the scrollar invariants of certaind-gonal curves (e.g. Castelnuovo curves and bielliptic curves) by using appropriate plane models. Ford=4 andg(C)≥10, we show that those invariants discriminate bielliptic curves among tetragonal ones.

Projective surfaces with bi-elliptic hyperplane sections

We study projective surfaces X which have a bi-elliptic curve (i.e. 2∶1 covering of an elliptic curve) among their hyperplane sections . We give a complete characterization of those surfaces when their degree d is d≥17 (only conic bundles and scrolls if d≥19, three possible exception otherwise) and when d≤8. A conjecture is given for the remaining cases. The main tool we use is the study of the adjunction mapping on X.

The secant line variety to the varieties of reducible plane curves

Let