Évènements

Smooth complex Fano varieties form a central class of projective varieties in algebraic geometry, whose classification is currently complete only in dimensions up to three. The Lefschetz defect is an invariant that has proved to offer an effective perspective in the study of smooth complex Fano varieties in arbitrary dimension. Recent breakthroughs show that when the Lefschetz defect is greater than two, one obtains strong restrictions on the geometry of the variety. In this talk, I focus on smooth Fano varieties with Lefschetz defect equal to two that arise from a specific construction introduced by C. Casagrande and S. Druel, together with some natural variants. We show that most Fano threefolds with defect two can be described via this construction. Moreover, in dimension four we complete the classification of all Fano varieties with defect two obtained in this way, resulting in a total of 173 distinct families.

In this talk we will discuss the connection between combinatorial properties of minimally self-intersecting curves on a surface S and the geometric behaviour of geodesics on S when S is endowed with a Riemannian metric. In particular, we will explain the interplay between a smoothing, which is a type of surgery on a curve that resolves a self-intersection, and k-systoles, which are shortest geodesics having at least k self-intersections, and we will present some results that partially elucidate this interplay. There will be lots of pictures. Based on joint work with Max Neumann-Coto.