Cones of lines having high contact with general hypersurfaces and applications

Given a smooth hypersurface X⊂P^{n+1} of degree d⩾2, we study the cones V^h_p⊂P^{n+1} swept out by lines having contact order h⩾2 at a point p∈X. In particular, we prove that if X is general, then for any p∈X and 2⩽h⩽min{n+1,d}, the cone V^h_p has dimension exactly n+2−h. Moreover, when X is a very general hypersurface of degree d⩾2n+2, we describe the relation between the cones V^h_p and the degree of irrationality of k-dimensional subvarieties of X passing through a general point of X. As an application, we give some bounds on the least degree of irrationality of k-dimensional subvarieties of X passing through a general point of X, and we prove that the connecting gonality of X satisfies suitable inequalities