Cones of lines having high contact with general hypersurfaces and applications

Given a smooth hypersurface $$X\subset \mathbb {P}^{n+1}$$ of degree $$d\geqslant 2$$, we study the cones $$V^h_p\subset \mathbb {P}^{n+1}$$ swept out by lines having contact order $$h\geqslant 2$$ at a point $$p\in X$$. In particular, we prove that if X is general, then for any $$p\in X$$ and $$2 \leqslant h\leqslant \min \lbrace n+1,d\rbrace$$, the cone $$V^h_p$$ has dimension exactly $$n+2-h$$. Moreover, when X is a very general hypersurface of degree $$d\geqslant 2n+2$$, 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 $$d-\left\lfloor \frac{\sqrt {16n+25}-3}{2}\right\rfloor \leqslant \operatorname{conn.gon}(X)\leqslant d-\left\lfloor \frac{\sqrt {8n+1}+1}{2}\right\rfloor$$.