We show that the arithmetic depth of the projection Xp of a rational normal scroll X̃ ⊂ P K from a point p ∈ P K \X̃ can be expressed in terms of the rank of the matrix M ′(p), where M ′ is the matrix of linear forms whose 3× 3 minors define the secant variety of X̃.

This article consists of three parts that are largely independent of one another. The first part deals with the projective geometry of homogeneous varieties, in particular their secant and tangential varieties. It culminates with an elementary construction of the compact Hermitian symmetric spaces and the closed orbits in the projectivization of the adjoint representation of a simple Lie algebr...

Each embedded product space PG(n, q)×PG(n, q) in an (n2 +n− 1)-dimensional projective space is obtained by projecting the Segre variety Sn,n,q from an n-subspace δ skew with its first secant variety. On the other hand, when δ is skew with the (n − 1)-th secant variety, it determines a semifield of order qn+1 whose center contains Fq. A relationship arises between a particular class of embedding...

A set without tangents in PG(2, q) is a set of points S such that no line meets S in exactly one point. An exterior set of a conic C is a set of points E such that all secant lines of E are external lines of C. In this paper, we first recall some known examples of sets without tangents and describe them in terms of determined directions of an affine pointset. We show that the smallest sets with...

Many problems in transportation can be formulated as fixed point problems, or equivalently as a systems of nonlinear equations, as for example multi-user equilibrium problems, planning problems, and consistent anticipatory route guidance generation. These fixed point problems are often characterized by the fact that there is no analytical form of the objective function, the computational cost o...

Young flattenings, introduced by Landsberg and Ottaviani, give determinantal equations for secant varieties and provide lower bounds for border ranks of tensors. We find special monomial-optimal Young flattenings that provide the best possible lower bound for all monomials up to degree 6. For degree 7 and higher these flattenings no longer suffice for all monomials. To overcome this problem we ...

We study the relationship between the equations defining a projective variety and properties of its secant varieties. In particular, we use information about the syzygies among the defining equations to derive smoothness and normality statements about SecX and also to obtain information about linear systems on the blow up of projective space along a variety X. We use these results to geometrica...

Let X ⊂ P be a generically reduced projective scheme. A fundamental goal in computational algebraic geometry is to compute information about X even when defining equations for X are not known. We use numerical algebraic geometry to develop a test for deciding if X is arithmetically Gorenstein and apply it to three secant varieties.

In this paper we give two explicit relations among 1-cycles modulo rational equivalence on a smooth cubic hypersurface X. Such a relation is given in terms of a (pair of) curve(s) and its secant lines. As the first application, we reprove Paranjape’s theorem that CH1(X) is always generated by lines and that it is isomorphic to Z if the dimension of X is at least 5. Another application is to the...

Throughout this paper we will assume that q is even. We will be considering ovoids of PG(3, q) which contain a conic as a plane section by using different representations of the classical generalized quadrangle of order q. An oŠal / of PG(2, q) is a set of q­1 points of PG(2, q), with no three collinear. Let F be a line of PG(2, q) ; then F is incident with zero, one or two points of / and is a...