The purpose of this paper is to relate the variety parameterizing completely decomposable homogeneous polynomials of degree d in n+1 variables on an algebraically closed field, called Splitd(P ), with the Grassmannian of n−1 dimensional projective subspaces of P. We compute the dimension of some secant varieties to Splitd(P ) and find a counterexample to a conjecture that wanted its dimension r...

For a smooth curve of genus g embedded by a line bundle of degree at least 2g + 3 we show that the ideal sheaf of the secant variety is 5-regular. This bound is sharp with respect to both the degree of the embedding and the bound on the regularity. Further, we show that the secant variety is projectively normal for the generic embedding of degree at least 2g + 3. AMS Subject Classification (200...

Let Xm,n be the Segre-Veronese variety P m ×P embedded by the morphism given by O(1, 2). In this paper, we provide two functions s(m, n) ≤ s(m, n) such that the s secant variety of Xm,n has the expected dimension if s ≤ s(m,n) or s(m, n) ≤ s. We also present a conjecturally complete list of defective secant varieties of such Segre-Veronese varieties.

The monotone secant conjecture posits a rich class of polynomial systems, all of whose solutions are real. These systems come from the Schubert calculus on flag manifolds, and the monotone secant conjecture is a compelling generalization of the Shapiro conjecture for Grassmannians (Theorem of Mukhin, Tarasov, and Varchenko). We present some theoretical evidence for this conjecture, as well as c...

We determine normal forms and ranks of tensors of border rank at most three. We present a differential-geometric analysis of limits of secant planes in a more general context. In particular there are at most four types of points on limiting trisecant planes for cominuscule varieties such as Grassmannians. We also show that the singular locus of the secant varieties σr(Seg(P × P × P)) has codime...

We formulate the Secant Conjecture, which is a generalization of the Shapiro Conjecture for Grassmannians. It asserts that an intersection of Schubert varieties in a Grassmannian is transverse with all points real if the flags defining the Schubert varieties are secant along disjoint intervals of a rational normal curve. We present theoretical evidence for this conjecture as well as computation...

A simple secant-based fast gradient method is developed for problems whose objective function is convex and well-defined. The proposed algorithm extends the classical Nesterov gradient method by updating the estimate-sequence parameter with secant information whenever possible. This is achieved by imposing a secant condition on the choice of search point. Furthermore, the proposed algorithm emb...