Cones of lines having high contact with general hypersurfaces and applications

Abstract

Given a smooth hypersurface $X\subset {\mathbb{P}}^{n+1}$ of degree $d⩾2$, we study the cones $V_p^h\subset {\mathbb{P}}^{n+1}$ swept out by lines having contact order $h⩾2$ at a point $p\in X$. In particular, we prove that if $X$ is general, then for any $p\in X$ and $2⩽h⩽\min \{n+1,d\}$, the cone $V_p^h$ 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 $V_p^h$ 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 $d-\left\lfloor\frac{\sqrt{16n+25}-3}{2}\right\rfloor\leqslant\conngon(X)\leqslant d-\left\lfloor\frac{\sqrt{8n+1}+1}{2}\right\rfloor$.

arXiv comment15 pages; v2: minor changes