Fano varieties : RAtional Curves, Arithmetic, Sections over Surfaces and Obstructions – FRACASSO

The main goal of the project is to study a recent higher version of rational connectedness on Fano varieties, rational simple connectedness, and its consequences on the arithmetic of rational points.
This notion involves rational connectedness of moduli spaces of rational curves. The main motivation for the project comes from the central role that Fano varieties have in the context of birational classification of algebraic varieties.
Rational simple connectedness is a generalisation of rational connectedness, where rational curves are replaced by surfaces. This can be seen as an algebro-geometric analogue of simple connectedness in topology and has relevant consequences on the structure of Fano fibrations over surfaces, namely on the existence of rational sections.
In order to understand which Fano varieties are rationally simply connected, we focus on several aspects, beyond the geometry of moduli spaces of rational curves on Fano varieties: the positivity of the second Chern character, the existence of twisting surfaces on Fano varieties and the geometry of rational points.