Mori fibre spaces, and more on the intermediate Jacobian of X_10

In the first talk we briefly recall the definition of a Mori fiber space. Then,
we show that for a Fano variety to be a general fibre of a Mori fibre space is
a rather restrictive condition. More precisely, we will give two criteria (one
sufficient and one necessary) for a Q-factorial Fano variety with terminal singularities
to be realised as a fibre of a Mori fibre space. These criteria give a
characterisation of rigid Fano varieties arising as a general fibre of a Mori fibre
space. We apply our criteria to Fano varieties of dimension at most three and
to rational homogeneous spaces.

In the second talk we continue our discussion of the Fano threefolds X10
following the work of Debarre-Iliev-Manivel and the more recent work of
Debarre-Kuznetzov. Our aim is to understand the (two-dimensional) fibres
of the intermediate Jacobian map from the moduli of Fano threefolds with Picard
rank one, index one and degree 10 to the moduli of principally polarized
ten-dimensional abelian varieties.