Restrictions on rational surfaces lying in very general hypersurfaces

TL;DR

This paper investigates the presence of rational surfaces in very general Fano hypersurfaces, demonstrating that for degrees close to the dimension, such hypersurfaces contain no maps from certain families of rational surfaces, impacting their unirationality.

Contribution

It establishes new restrictions on rational surfaces in very general hypersurfaces, showing the absence of certain maps and rational curves, advancing understanding of their geometric properties.

Findings

No maps from fixed families of rational surfaces in high-degree hypersurfaces

Rational curves in the space must meet the boundary in these hypersurfaces

Hypersurfaces admit no maps from surfaces with specific intersection properties

Abstract

We study rational surfaces on very general Fano hypersurfaces in $\mathbb{P}^n$, with an eye toward unirationality. We prove that given any fixed family of rational surfaces, a very general hypersurface of degree $d$ sufficiently close to $n$ and $n$ sufficiently large will admit no maps from surfaces in that family. In particular, this shows that for such hypersurfaces, any rational curve in the space of rational curves must meet the boundary. We also prove that for any fixed ratio $\alpha$, a very general hypersurface in $\mathbb{P}^n$ of degree $d$ sufficiently close to $n$ will admit no maps from a surface satisfying $H^2 \geq \alpha HK$, where $H$ is the pullback of the hyperplane class from $\mathbb{P}^n$ and $K$ is the canonical bundle on the surface.