A note on the birational geometry of tropical line bundles

Given a closed subvariety Y of a n-dimensional torus, we study how the tropical line bundles of Trop(Y) can be induced by line bundles living on a tropical compactification of Y in a toric variety, following the construction of Jenia Tevelev. We then consider the general structure with respect to the Zariski--Riemann space.Comment: Version

Ruled Fano fivefolds of index two

We classify Fano fivefolds of index two which are projectivization of rank two vector bundles over four dimensional manifolds.Comment: 30 page

Extremal rays of non-integral LL-length

Let $X$ be a smooth complex projective variety and let $L$ be a line bundle on it. We describe the structure of the pre-polarized manifold $(X,L)$ for non integral values of the invariant $\tau_L(R):=-K_X\cdot\Gamma/(L \cdot \Gamma)$, where $\Gamma$ is a minimal curve of an extremal ray $R:=\mathbb R_+[\Gamma]$ on $X$ such that $L \cdot R>0$

Extremal rays of non-integral LL-length

Let $X$ be a smooth complex projective variety and let $L$ be a line bundle on it. We describe the structure of the pre-polarized manifold $(X,L)$ for non integral values of the invariant $\tau_L(R):=-K_X\cdot\Gamma/(L \cdot \Gamma)$, where $\Gamma$ is a minimal curve of an extremal ray $R:=\mathbb R_+[\Gamma]$ on $X$ such that $L \cdot R>0$.Comment: 13 page

On Fano manifolds with an unsplit dominating family of rational curves

We study Fano manifolds $X$ admitting an unsplit dominating family of rational curves and we prove that the Generalized Mukai Conjecture holds if $X$ has pseudoindex $i_X = (\dim X)/3$ or dimension $\dim X=6$. We also show that this conjecture is true for all Fano manifolds with $i_X > (\dim X)/3$.Comment: 11 pages. arXiv admin note: substantial text overlap with arXiv:0905.438

Manifolds covered by lines and extremal rays

Let $X$ be a smooth complex projective variety and let H \in \pic(X) be an ample line bundle. Assume that $X$ is covered by rational curves with degree one with respect to $H$ and with anticanonical degree greater than or equal to $(\dim X -1)/2$. We prove that there is a covering family of such curves whose numerical class spans an extremal ray in the cone of curves \cone(X).Comment: Major revision, to appear in Canadian Mathematical Bulleti