Rational weighted projective hypersurfaces

TL;DR

This paper develops new methods to construct rational weighted hypersurfaces of higher degree, expanding the known examples and confirming the existence of rational Fano hypersurfaces in all dimensions at least six.

Contribution

It introduces rationality constructions for weighted hypersurfaces of higher degree, answering an open question about the existence of rational Fano hypersurfaces in all dimensions ≥6.

Findings

New rational examples of weighted hypersurfaces in various dimensions.

Confirmed the existence of very general terminal Fano rational weighted hypersurfaces in all dimensions ≥6.

Extended the understanding of rationality in weighted projective hypersurfaces.

Abstract

A very general hypersurface of dimension $n$ and degree $d$ in complex projective space is rational if $d \leq 2$, but is expected to be irrational for all $n, d \geq 3$. Hypersurfaces in weighted projective space with degree small relative to the weights are likewise rational. In this paper, we introduce rationality constructions for weighted hypersurfaces of higher degree that provide many new rational examples over any field. We answer in the affirmative a question of T. Okada about the existence of very general terminal Fano rational weighted hypersurfaces in all dimensions $n \geq 6$.