Cones of lines having high contact with general hypersurfaces and applications

TL;DR

This paper investigates the geometry of cones of lines with high contact order on smooth hypersurfaces, establishing their dimensions and relations to irrationality measures, with applications to bounds on gonality.

Contribution

It provides explicit descriptions of cones of lines with high contact order on general hypersurfaces and relates these to the irrationality and gonality of subvarieties.

Findings

Cones $V^h_p$ have dimension exactly $n+2-h$ for general hypersurfaces.

Bounds on the least degree of irrationality of subvarieties passing through a general point.

Bounds on the connecting gonality of the hypersurface.

Abstract

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 $2 \leqslant h\leqslant \min\{ n+1,d\}$, the cone $V^h_p$ has dimension exactly $n+2-h$. Moreover, when $X$ is a very general hypersurface of degree $d\geqslant 2n+2$, 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 $d-\left\lfloor\frac{\sqrt{16n+25}-3}{2}\right\rfloor\leqslant\conngon(X)\leqslant d-\left\lfloor\frac{\sqrt{8n+1}+1}{2}\right\rfloor$.