Related papers: Real Rank Two Geometry

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…

The $X$-rank of a point $p$ in projective space is the minimal number of points of an algebraic variety $X$ whose linear span contains $p$. This notion is naturally submultiplicative under tensor product. We study geometric conditions that…

This paper is devoted to the description of complex finite-dimensional algebras of level two. We obtain the classification of algebras of level two in the varieties of Jordan, Lie and associative algebras.

We consider higher secant varieties to Veronese varieties. Most points on the r-th secant variety are represented by a finite scheme of length r contained in the Veronese variety --- in fact, for generic point, it is just a union of r…

We study typical ranks with respect to a real variety $X$. Examples of such are tensor rank ($X$ is the Segre variety) and symmetric tensor rank ($X$ is the Veronese variety). We show that any rank between the minimal typical rank and the…

For a given projective variety $X$, the high rank loci are the closures of the sets of points whose $X$-rank is higher than the generic one. We show examples of strict inclusion between two consecutive high rank loci. Our first example is…

We study orthogonal decompositions of symmetric and ordinary tensors using methods from linear algebra. For the field of real numbers we show that the sets of decomposable tensors can be defined be equations of degree 2. This gives a new…

We study the case of a real homogeneous polynomial $P$ whose minimal real and complex decompositions in terms of powers of linear forms are different. We prove that, if the sum of the complex and the real ranks of $P$ is at most $…

We study the algebraic features of covariant tensors of valence four containing two blocks of skew indices. After a rather general treatment, we specialize ourselves to four-dimensional spacetimes and discuss several complementary aspects…

A linear system of real quadratic forms defines a real projective variety. The real non-singular locus of this variety (more precisely of the underlying scheme) has a highly connected double cover as long as each non-zero form in the system…

We study the local differential geometry of varieties $X^n\subset \Bbb C\Bbb P^{n+a}$ with degenerate secant and tangential varieties. We show that the second fundamental form of a smooth variety with degenerate tangential variety is…

A Laurent polynomial in two variables is tempered if its edge polynomials are cyclotomic. Variation of coefficients leads to a family of smooth complete genus $g$ curves carrying a canonical algebraic $K_2$-class over a $g$-dimensional base…

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…

In the present article, we investigate the topology of real toric varieties, especially those whose torus is not split over the field of real numbers. We describe some canonical fibrations associated to their real loci. Then, we establish…

A new approach to the algebraic classification of second order symmetric tensors in 5-dimensional space-times is presented. The possible Segre types for a symmetric two-tensor are found. A set of canonical forms for each Segre type is…

The bad locus and the rude locus of an ample and base point free linear system on a smooth complex projective variety are introduced and studied. The bad locus is defined as the set of points that force divisors through them to be…

We study an upper bound of ranks of $n$-tensors with size $2\times\cdots\times2$ over the complex and real number field. We characterize a $2\times 2\times 2$ tensor with rank 3 by using the Cayley's hyperdeterminant and some function. Then…

Though algebraic geometry over $\mathbb C$ is often used to describe the closure of the tensors of a given size and complex rank, this variety includes tensors of both smaller and larger rank. Here we focus on the $n\times n\times n$…

We construct wonderful compactifications of the spaces of linear maps, and symmetric linear maps of a given rank as blow-ups of secant varieties of Segre and Veronese varieties. Furthermore, we investigate their birational geometry and…

We study the semialgebraic structure of $D_r$, the set of nonnegative tensors of nonnegative rank not more than $r$, and use the results to infer various properties of nonnegative tensor rank. We determine all nonnegative typical ranks for…