On projective varieties 3-covered by curves of degree d

Abstract: 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).