On projective varieties 3-covered by curves of degree d

I will present some results concerning projective varieties X that are 3-connected by irreducible curves of a fixed degree d. We will restrict to the study of the varieties of this type spanning a linear space of maximal possible dimension. Under this extremality assumption, we will prove that X is rational, that there exists a unique 3-covering familly F of curves on X and that the general element of F is a rational normal curve. Then we will provide a complete classification of such extremal X when d>3. If time allows, I plan to consider also the case d=3 of extremal projective varieties X that are 3-covered by twisted cubics. I will show that this case is more interesting than the general one and that there are equivalences between -- projective equivalence classes of such varieties X; -- Cremona transformations of bidegree (2,2) (up to linear equivalence); -- rank 3 Jordan algebras (up to isotopy); (The first part of the talk is based on a joint work with J.M. Trépreau, the second one on a collaboration with F. Russo).