Cinzia Casagrande

Let X be a smooth, complex Fano variety, D a prime divisor in X, and 
set c(D):=dim ker(r:H2(X,R)->H2(D,R)), where r is the natural 
restriction map. It is a special property of Fano manifolds that the 
presence of a prime divisor D with large c(D) has consequences on the 
geometry of X.
More precisely, we define c_X:=max{c(D)|D is a prime divisor in X}.
Then c_X\leq 8, and if c_X is at least 2, then we get some special 
properties of X. We will explain this result, which relies on a 
construction in birational geometry; then we will focus on the case 
c_X=2.