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Let $X$ be a complex Fano manifold of arbitrary dimension, and $D$ a prime divisor in $X$. We consider the image ${𝒩}_1(D,X)$ of ${𝒩}_1\left(D\right)$ in ${𝒩}_1\left(X\right)$ under the natural push-forward of $1$-cycles. We show that ${\rho }_X-{\rho }_D\le codim{𝒩}_1(D,X)\le 8$. Moreover if $codim{𝒩}_1(D,X)\ge 3$, then either $X\cong S\times T$ where $S$ is a Del Pezzo surface, or $codim{𝒩}_1(D,X)=3$ and $X$ has a fibration in Del Pezzo surfaces onto a Fano manifold $T$ such that ${\rho }_X-{\rho }_T=4$.

This is the text of a talk given at the XVII Convegno dell’Unione Matematica Italiana held at Milano, September 8-13, 2003. I would like to thank Angelo Lopez and Ciro Ciliberto for the kind invitation to the conference. I survey some numerical conjectures and theorems concerning relations between the index, the pseudo-index and the Picard number of a Fano variety. The results I refer to are contained in the paper [3], wrote in collaboration with Bonavero, Debarre and Druel.

Let $X$ be a Gorenstein, $\mathbb{Q}$-factorial, toric Fano variety. We prove two conjectures on the maximal Picard number of $X$ in terms of its dimension and its pseudo-index, and characterize the boundary cases. Equivalently, we determine the maximal number of vertices of a simplicial reflexive polytope.

Given a covering family $V$ of effective 1-cycles on a complex projective variety $X$, we find conditions allowing one to construct a geometric quotient $q:X\to Y$, with $q$ regular on the whole of $X$, such that every fiber of $q$ is an equivalence class for the equivalence relation naturally defined by $V$. Among other results, we show that on a normal and $\mathbb{Q}$-factorial projective variety $X$ with canonical singularities and $dimX\le 4$, every covering and quasi-unsplit family $V$ of rational curves generates a geometric extremal...