On the secant varieties to the osculating variety of a Veronese surface

In this paper we study the k-th osculating variety of the order d Veronese embedding of P ⁿ . In particular, for k=n=2 we show that the corresponding secant varieties have the expected dimension except in one case.     

Chordal varieties of Veronese varieties and catalecticant matrices

It is proved that the chordal variety of the Veronese variety is projectively normal, arithmetically caulay, and its homogeneous ideal is generated by the 3×3 minors of two catalecticant matrices. Results are generalized to the catalecticant varieties Gor_≤(T) witht ₁=2. Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 56. Algebraic Geometry-9, 1998.     

Characterizations of Veronese and Segre varieties

We survey the most important characterizations of quadric Veroneseans and Segre varieties of the last thirty years, including some very recent results.     

An asymptotic bound for secant varieties of Segre varieties

This paper studies the defectivity of secant varieties of Segre varieties. We prove that there exists an asymptotic lower estimate for the greater non-defective secant variety (without filling the ambient space) of any given Segre variety. In particular, we prove that the ratio between the greater non-defective secant variety of a Segre variety and its expected rank is lower bounded by a value depending just on the number of factors of the Segre variety. Moreover, in the final section, we present some results obtained by explicit computation, proving the non-defectivity of all the secant varieties of Segre varieties of the shape $(\mathbb{P }^{n})^4$ , with $2 \le n\le 10$ , except at most $\sigma _{199}((\mathbb{P }^8)^4)$ and $\sigma _{357}((\mathbb{P }^{10})^4)$ .     

A note on Veronese varieties

We show that for every primep, there is a class of Veronese varieties which are set-theoretic complete intersections if and only if the ground field has characteristicp. Partially supported by PRIN Algebra Commutativa e Computazionale, Italian Ministry of Education, University and Research.     

Higher order secant varieties and blown up varieties

Let π:X→Y be the blowing up of the projective varietyY at s general points. Here we study the higher order secant varieties of the linearly normal embeddings ofX andY into projective spaces. We give conditions on the embedding ofY which imply that the firstt secant varieties of a related embedding ofX have the expected dimension.     

Higher order secant varieties and blown up varieties

Let π:X→Y be the blowing up of the projective varietyY at s general points. Here we study the higher order secant varieties of the linearly normal embeddings ofX andY into projective spaces. We give conditions on the embedding ofY which imply that the firstt secant varieties of a related embedding ofX have the expected dimension.     

Osculating spaces to secant varieties

We generalize the classical Terracini’s Lemma to higher order osculating spaces to secant varieties. As an application, we address with the so-called Horace method the case of thed-Veronese embedding of the projective 3-space. This research is part of the T.A.S.C.A. project of I.N.d.A.M., supported by P.A.T. (Trento) and M.I.U.R. (Italy).     

Distinguishing secant from cactus varieties

Foundations of Computational Mathematics - Cactus varieties are a generalization of secant varieties. They are defined using linear spans of arbitrary finite schemes of bounded length, while secant...     

On the dimensions of secant varieties of Segre-Veronese varieties

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 sth Segre–Veronese varieties for small s. In the final section, we propose a conjecture about defective two-factor Segre–Veronese varieties.     

On the ideals and singularities of secant varieties of Segre varieties

We find minimal generators for the ideals of secant varietiesof Segre varieties in the cases of _k(¹ x ⁿ x ^m) for all k, n,m, ₂(ⁿ x ^m x ^p x ^r) for all n, m, p, r (GSS conjecture for fourfactors), and ₃(ⁿ x ^m x ^p) for all n, m, p and prove they arenormal with rational singularities in the first case and arithmeticallyCohen–Macaulay in the second two cases.     

On secant dimensions and identifiability of flag varieties

We investigate the secant dimensions and the identifiability of flag varieties parametrizing flags of subspaces of a fixed vector space. We give numerical conditions ensuring that secant varieties of flag varieties have the expected dimension, and that a general point on these secant varieties is identifiable.     

Asymptotic syzygies of secant varieties of curves

We prove that the minimal free resolution of the secant variety of a curve is asymptotically pure. As a corollary, we show that the Betti numbers of converge to a normal distribution.     

Singular Poisson-Kähler geometry of Scorza varieties and their secant varieties

Each Scorza variety and its secant varieties in the ambient projective space are identified, in the realm of singular Poisson-Kähler geometry, in terms of projectivizations of holomorphic nilpotent orbits in suitable Lie algebras of hermitian type, the holomorphic nilpotent orbits, in turn, being affine varieties. The ambient projective space acquires an exotic Kähler structure, the closed stratum being the Scorza variety and the closures of the higher strata its secant varieties. In this fashion, the secant varieties become exotic projective varieties. In the rank 3 case, the four regular Scorza varieties coincide with the four critical Severi varieties. In the standard cases, the Scorza varieties and their secant varieties arise also via Kähler reduction. An interpretation in terms of constrained mechanical systems is included.