Cinzia Casagrande

Mukai has realised the blow-up X of $\mathbb{P}^4$ in 8 general points as a moduli space of vector bundles on a degree 1 del Pezzo surface S. With the same construction, we associate to S a (smooth) Fano 4-fold Y, which is isomorphic to X in codimension 1. We will explain the interplay between S, X, and Y, which allows to describe many geometrical properties of X and Y, in particular in relation to birational geometry. For instance, we can describe the relevant cones of divisors and the Mori chamber decomposition of these 4-folds. The Fano 4-fold Y has an interesting geometry; it has b_2(Y)=9, (K_Y)=13, and dim|-K_Y|=5. The linear system |-K_Y| has non-empty base locus, while |-2K_Y| is base-point-free. The second cohomology group H^2(Y,Z) has a lattice structure such that the ortogonal of the canonical class is an E_8-lattice; the associated Weyl group fixes the cones of effective, movable and nef divisors, and permutes the chambers of the Mori chamber decomposition.

This is a joint work with Giulio Codogni and Andrea Fanelli.