| According to Gushel and Mukai, most Fano varieties of degree 10, dimension $n$, and coindex 3 (index $n-2$) are obtained as linear sections of the intersection of the Grassmannian $G(2,5)$ in its Plücker embedding with a quadric. In particular, $n$ can only be 3, 4, or 5; they depend on 22, 24, and 25 parameters respectively. We will study the geometry of these varieties, their links with the double EPW-sextics studied by O'Grady, and their various period maps. We work over the complex numbers, and this is joint work with A. Iliev and L. Manivel. |