Cones of lines having high contact with general hypersurfaces and applications

Given a smooth hypersurface X⊂Pn+1$X\subset \mathbb {P}^{n+1}$ of degree d⩾2$d\geqslant 2$, we study the cones Vph⊂Pn+1$V^h_p\subset \mathbb {P}^{n+1}$ swept out by lines having contact order h⩾2$h\geqslant 2$ at a point p∈X$p\in X$. In particular, we prove that if X is general, then for any p∈X$p\in X$ and 2⩽h⩽min{n+1,d}$2 \leqslant h\leqslant \min \lbrace n+1,d\rbrace$, the cone Vph$V^h_p$ has dimension exactly n+2−h$n+2-h$. Moreover, when X is a very general hypersurface of degree d⩾2n+2$d\geqslant 2n+2$, we describe the relation between the cones Vph$V^h_p$ 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−16n+25−32⩽conn.gon(X)⩽d−8n+1+12$d-\left\lfloor \frac{\sqrt {16n+25}-3}{2}\right\rfloor \leqslant \operatorname{conn.gon}(X)\leqslant d-\left\lfloor \frac{\sqrt {8n+1}+1}{2}\right\rfloor$.