An FGM (Fano-Gushel-Mukai) variety is a Fano variety with Picard number 1 coindex 2 and degree 10 (with respect to the generator of the Picard group). These varieties were classified in works of Gushel and Mukai who showed that each of those can be obtained either as an intersection of Gr(2,5) with a quadric, or a double cover of Gr(2,5) ramified in a quadric, or a linear section of one of these. I will discuss a completely new proof of this result based on the use of the excess normal bundle. I will also describe the geometry of FGM varieties, their relation to EPW (Eisenbud-Popescu-Walter) sextics, the period map, and the birational transformations between these varieties.
This is a joint work in progress with Olivier Debarre.