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Abstract. This paper studies the dimension of secant varieties to Segre varieties. The problem is cast both in the setting of tensor algebra and in the setting of algebraic geometry. An inductive procedure is built around the ideas of successive specializations of points and projections. This reduces the calculation of the dimension of the secant variety in a high dimensional case to a sequence of calculations of partial secant varieties in low dimensional cases. As applications of the technique: We give a complete classification of defective p-secant varieties to Segre varieties for p ≤ 6. We generalize a theorem of Catalisano-Geramita-Gimigliano on non-defectivity of tensor powers of Pn. We determine the set of p for which unbalanced Segre varieties have defective p-secant varieties. In addition, we completely describe the dimensions of the secant varieties to the deficient Segre varieties P1 × P1 × Pn × Pn and P2 × P3 × P3. In the final section we propose a series of conjectures about defective Segre varieties.

Let Gr(k, n) be the Plücker embedding of the Grassmann variety of projective k-planes in P n . For a projective variety X, let σ s (X) denote the variety of its secant (s − 1)-planes. More precisely, σ s (X) denotes the Zariski closure of the union of linear spans of s-tuples of points lying on X. We exhibit two functions s 0 (n) ≤ s 1 (n) such that σ s (Gr(2, n)) has the expected dimension whenever n ≥ 9 and either s ≤ s 0 (n) or s 1 (n) ≤ s. Both s 0 (n) and s 1 (n) are asymptotic to n 2 18 . This yields, asymptotically, the typical rank of an element of 3 C n+1 . Finally, we classify all defective σ s (Gr(k, n)) for s ≤ 6 and provide geometric arguments underlying each defective case.This leads to the following proposition: Proposition 4.3. Let X = Gr(2, n) with n ≥ 11. Let V = K n+1 , and let L and M be general codimension six subspaces of V . Let L (resp. M) be the Grassmann variety of 3-planes in L (resp. M ). Then (i) The system of hyperplanes in P( 3 V ) which contain L∪M and which contain the tangent spaces at 6n−49 9 general points on L, at 6n−49 9 general points on M, and at 4 general points on X has the expected dimension 36(n − 6) − 36 6n−49 9 − 4(3n − 5), which is ⎧ ⎨ ⎩ 20 if n = 0 (mod 3), 8 if n = 1 (mod 3), 32 if n = 2 (mod 3). (ii) There are no hyperplanes in P( 3 V ) which contain L ∪ M and which contain the tangent spaces at 6n−49 9 general points on L, at 6n−49 9 general points on M, and at 4 general points on X. Proof. We will let {p i } denote the 6n−49 9 (resp. 6n−49 9 ) general points on L, {q i } denotes the 6n−49 9 (resp. 6n−49 9 ) general points on M and {r i } denotes the four general points on X. The proof is by a 6-step induction Licensed to Univ of Mass Amherst. Prepared on Sun Jul 5

Abstract. The solution set V of a polynomial system, i.e., the set of common zeroes of a set of multivariate polynomials with complex coefficients, may contain several components, e.g., points, curves, surfaces, etc. Each component has attached to it a number of quantities, one of which is its dimension. Given a numerical approximation to a point p on the set V , this article presents an efficient algorithm to compute the maximum dimension of the irreducible components of V which pass through p, i.e., a local dimension test. Such a test is a crucial element in the homotopy-based numerical irreducible decomposition algorithms of Sommese, Verschelde, and Wampler.This article presents computational evidence to illustrate that the use of this new algorithm greatly reduces the cost of so-called "junk-point filtering," previously a significant bottleneck in the computation of a numerical irreducible decomposition. For moderate size examples, this results in well over an order of magnitude improvement in the computation of a numerical irreducible decomposition. As the computation of a numerical irreducible decomposition is a fundamental backbone operation, gains in efficiency in the irreducible decomposition algorithm carry over to the many computations which require this decomposition as an initial step. Another feature of a local dimension test is that one can now compute the irreducible components in a prescribed dimension without first computing the numerical irreducible decomposition of all higher dimensions. For example, one may compute the isolated solutions of a polynomial system without having to carry out the full numerical irreducible decomposition.

Communicated by A. J. SommeseWe present a method to compute the degrees of the Segre classes of a subscheme of complex projective space. The method is based on generic residuation and intersection theory. We provide a symbolic implementation using the software system Macaulay2 and a numerical implementation using the software package Bertini.