Dominant rational maps from a very general hypersurface

Abstract: There has been recent interest in studying measures of irrationality for hypersurfaces $X$ of high degree in $\mathbb P^{n+1}$. The degree of irrationality of $X$ is defined as the minimal degree of dominant rational maps from $X$ to $\mathbb P^n$. It is known that the degree of irrationality of $X$ is $d-1$ if $X$ is a very general hypersurface of degree $d\ge 2n+1$. In this talk, from a different point of view we will discuss dominant rational maps of finite degree from a very general hypersurface $X$ of degree $d\ge n+3$ in $\mathbb P^{n+1}$ to any smooth projective variety $Z$. The finite theorem states that these form a finite set, up to birational equivalence of $Z$, if $Z$ is a variety of general type. It is an interesting question to determine $Z$ when $Z$ is not birational to $X$. It is conjecturally expected that $Z$ is rationally connected if $X$ is a very general hypersurface of degree $d\ge n+3$. In this talk, I will present some partial results for this expectation. This talk combines the joint work with Gian Pietro Pirola and the joint work with De-Qi Zhang.

algebraic geometry

Audience: researchers in the topic

ZAG (Zoom Algebraic Geometry) seminar

Series comments: Description: ZAG seminar

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