Secant line

Abstract: We will start by giving a number of different definitions of dimension and we will try to indicate how they relate to one another. All of our definitions initially apply to an irreducible variety X; the dimension of an arbitrary variety will be defined to be the maximum of the dimensions of its irreducible components.

Abstract: The number of apparent double points of a smooth, irreducible projective variety $X$ of dimension $n$ in $\Proj^{2n+1}$ is the number of secant lines to $X$ passing through the general point of $\Proj^{2n+1}$. This classical notion dates back to Severi. In the present paper we classify smooth varieties of dimension at most three having one apparent double point. The techniques developed for this purpose allow to treat a wider class of projective varieties.

Abstract: Quadratic entry locus manifold of type $\delta$ $X\subset\mathbb P^N$ of dimension $n\geq 1$ are smooth projective varieties such that the locus described on $X$ by the points spanning secant lines passing through a general point of the secant variety $SX\subseteq\mathbb P^N$ is a smooth quadric hypersurface of dimension $\delta=2n+1-\dim(SX)$ equal to the secant defect of $X$. These manifolds appear widely and naturally among projective varieties having special geometric properties and/or extremal tangential behaviour. We prove that, letting $\delta=2r_X +1\geq 3$ or $\delta=2r_X+2$, then $2^{r_X}$ divides $n-\delta$. This is obtained by the study of the projective geometry of the Hilbert scheme $Y_x\subset \mathbb(T_x^*)$ of lines passing through a general point $x$ of $X$, allowing an inductive procedure. The Divisibility Property described above allows unitary and simple proofs of many results on $QEL$-manifolds such as the complete classification of those of type $\delta\geq n/2$, of Cremona transformation of type $(2,3)$, $(2,5)$. In particular we propose a new and very short proof of the fact that Severi varieties have dimension 2,4, 8 or 16 and also an almost self contained half page proof of their classification due to Zak.

Abstract: 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 $ \mathrm {CH}_1(X)$ is always generated by lines and that it is isomorphic to $ \mathbb{Z}$ if the dimension of $ X$ is at least 5. Another application is to the intermediate jacobian of a cubic threefold $ X$. To be more precise, we show that the intermediate jacobian of $ X$ is naturally isomorphic to the Prym-Tjurin variety constructed from the curve parameterizing all lines meeting a given rational curve on $ X$. The incidence correspondences play an important role in this study. We also give a description of the Abel-Jacobi map for 1-cycles in this setting. - See more at: http://www.ams.org/journals/jag/2014-23-03/S1056-3911-2014-00631-7/home.html#sthash.T8vmMhvP.dpuf