Related papers: Cremona linearizations of some classical varieties

We introduce the concise secant varieties, which are, informally speaking, modular partial desingularisations of secant varieties to Segre embeddings. More precisely, they are projective and birational to the abstract secant varieties, yet…

Going one step further in Zak's classification of Scorza varieties with secant defect equal to one, we characterize the Veronese embedding of $\P^n$ given by the complete linear system of quadrics and its smooth projections from a point as…

In the present paper we construct quadratic equations and linear syzygies for tangent varieties using 4-way tensors of linear forms and generalize this method to higher secant varieties of higher osculating varieties. Such equations extend…

We classify complex projective varieties of dimension $2r \geq 8$ swept out by a family of codimension two grassmannians of lines $\mathbb{G}(1,r)$. They are either fibrations onto normal surfaces such that the general fibers are isomorphic…

We characterize the double Veronese embedding of P^n as the only variety that, under certain general conditions, can be isomorphically projected from the Grassmannian of lines in P^{2n+1} to the Grassmannian of lines in P^{n+1}.

We prove that the moduli spaces of rational curves of degree at most $3$ in linear sections of the Grassmannian $Gr(2,5)$ are all rational varieties. We also study their compactifications and birational geometry.

We describe a computational proof that the fifth secant variety of the Segre product of five copies of the projective line is a codimension 2 complete intersection of equations of degree 6 and 16. Our computations rely on pseudo-randomness,…

This article is an expository account of the theory of twisted commutative algebras, which simply put, can be thought of as a theory for handling commutative algebras with large groups of linear symmetries. Examples include the coordinate…

In the sl\_n case, A. Berenstein and A. Zelevinsky studied the Sch\"{u}tzenberger involution in terms of Lusztig's canonical basis, [3]. We generalize their construction and formulas for any semisimple Lie algebra. We use for this the…

Let $X_{m,n}$ be the Segre-Veronese variety $\mathbb{P}^m \times \mathbb{P}^n$ embedded by the morphism given by $\mathcal{O}(1,2)$. In this paper, we provide two functions $\underline{s}(m,n)\le \bar{s}(m,n)$ such that the…

We classify the homogeneous polynomials in three variables whose toric polar linear system defines a Cremona transformation. This classification also includes, as a proper subset, the classification of toric surface patches from geometric…

Motivated by the study of the secant variety of the Segre-Veronese variety we propose a general framework to analyze properties of the secant varieties of toric embeddings of affine spaces defined by simplicial complexes. We prove that…

We determine the minimal generators of the ideal of the tangential variety of a Segre-Veronese variety, as well as the decomposition into irreducible GL-representations of its homogeneous coordinate ring. In the special case of a Segre…

We study the secant line variety of the Segre product of projective spaces using special cumulant coordinates adapted for secant varieties. We show that the secant variety is covered by open normal toric varieties. We prove that in cumulant…

Multigraded linear series generalize the classical morphism to the linear series of a basepoint-free line bundle on a scheme. We investigate the collection of the natural cornering morphisms into elementary bigraded linear series obtained…

A cheap method for constructing canonical models and complete moduli for complex projective varieties with a structure called "rational plurifibration" is given. A result about semistable reduction (whose nature is slightly different from…

Hyperplanes and hyperplane complements in the Segre product of partial linear spaces are investigated . The parallelism of such a complement is characterized in terms of the point-line incidence. Assumptions, under which the automorphisms…

The concept of a fully interlacing matrix of formal power series with real coefficients is introduced. This concept extends and strengthens that of an interlacing sequence of real-rooted polynomials with nonnegative coefficients, in the…

Secant defectivity of projective varieties is classically approached via dimensions of linear systems with multiple base points in general position. The latter can be studied via degenerations. We exploit a technique that allows some of the…

We survey results in algebraic complexity theory, focusing on matrix multiplication. Our goals are (i.) to show how open questions in algebraic complexity theory are naturally posed as questions in geometry and representation theory, (ii.)…