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.

In this paper we give a cohomological proof for the existence of a new family of smooth surfaces in P 4 . In fact, we show: Theorem 0.1. (i) Let X ⊂ P 4 be an elliptic conic bundle with degree d = 8 and sectional genus π = 5. Then the ideal sheaf of X arises as a cokernelwhere G is a rank-5 vector bundle on P 4 with Chern-classesG is isomorphic to the cohomology bundle of a monadwhere e 0 , . . . , e 4 is a basis of the underlying vector space V of P 4 . In particular, G is uniquely determined up to isomorphisms and coordinate transformations. 726H. Abo et al.(ii) Conversely, if G is the cohomology bundle of the monad (M ) as in (i), then G(1) is globally generated. Therefore the dependancy locus of four general sections of G(1) is a smooth surface X ⊂ P 4 . In fact, X is an elliptic conic bundle with invariants as above.For a geometric construction of our surfaces we refer to [Ra].The new surfaces are missing in a series of classification papers. First of all, they are falsely ruled out in the classification of smooth degree 8 surfaces in P 4 by Okonek [Ok2] and, independently, Ionescu [Io]. As a consequence, they are e.g. also missing in the first version of two papers which are concerned with the classification of conic bundles in P 4 [ES], [BR]. The correct result is:Let X ⊂ P 4 be a smooth surface ruled in conics. Then X is either rational or an elliptic conic bundle with d = 8 and π = 5. In the first case X is either a Del Pezzo surface of degree 4 or a Castelnuovo surface. This result is important in the context of adjunction theory [So], [SV], [VdV].Recall, that for a smooth surface X ⊂ P 4 the adjunction map is defined unless X is a plane or a scroll. If the adjunction map is defined, then it has a 2-dimensional image unless X is a Del Pezzo surface or a conic bundle. Therefore the classification of scrolls in P 4 [La], [Au] and the above result imply: Corollary 0.3. Let X ⊂ P 4 be a smooth surface of degree d ≥ 9. Then the adjunction map is defined and has a 2-dimensional image.The new family is one of a few known families of irregular smooth surfaces in P 4 . In fact, up to pullbacks via finite covers P 4 → P 4 , our surfaces are the first such surfaces which do not possess a Heisenberg symmetry (compare [ADHPR]). Moreover, they provide a counterexample to a conjecture of Ellingsrud and Peskine. According to this conjecture there should be no irregular m-ruled surface in P 4 for m ≥ 2.We first came across the elliptic conic bundles when studying a stable rank-3 vector bundle E on P 4 with Chern classes c 1 = 4, c 2 = 8 and c 3 = 8. E has been found by the stratification theoretical method of the third author (compare [Sa] for this method). The dependancy locus of two sections of E is a smooth surface of the desired type. In fact, E is a cokernelwhere G is the rank-5 vector bundle from Theorem 0.1.Elliptic conic bundles in P 4 727 Our paper is organized as follows: In Sect. 1 we review Beilinson's theorem [Bei] in the context of smooth surfaces in P 4 . In Sect. 2 we follow the cohomological approach ...

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

We prove a conjecture stated by Catalisano, Geramita, and Gimigliano in 2002, which claims that the secant varieties of tangential varieties to the d d th Veronese embedding of the projective n n -space P n \mathbb {P}^n have the expected dimension, modulo a few well-known exceptions. It is arguably the first complete result on the dimensions of secant varieties of a classically studied variety since the work of Alexander and Hirschowitz in 1995. As Bernardi, Catalisano, Gimigliano, and Idá demonstrated that the proof of this conjecture may be reduced to the case of cubics, i.e., d = 3 d=3 , the main contribution of this work is the resolution of this base case. The proposed proof proceeds by induction on the dimension n n of the projective space via a specialization argument. This reduces the proof to a large number of initial cases for the induction, which were settled using a computer-assisted proof. The individual base cases were computationally challenging problems. Indeed, the largest base case required us to deal with the tangential variety to the third Veronese embedding of P 79 \mathbb {P}^{79} in P 88559 \mathbb {P}^{88559} .

This paper explores the dimensions of higher secant varieties to Segre-Veronese varieties. The main goal of this paper is to introduce two different inductive techniques. These techniques enable one to reduce the computation of the dimension of the secant variety in a high dimensional case to the computation of the dimensions of secant varieties in low dimensional cases. As an application of these inductive approaches, we will prove non-defectivity of secant varieties of certain two-factor Segre-Veronese varieties. We also use these methods to give a complete classification of defective s-th Segre-Veronese varieties for small s. In the final section, we propose a conjecture about defective two-factor Segre-Veronese varieties.Comment: Revised version. To appear in Annali di Matematica Pura e Applicat