Rational weighted projective hypersurfaces

Abstract

A very general hypersurface of dimension $n$ and degree $d$ in complex projective space is rational if $d\le 2$, but is expected to be irrational for all $n,d\ge 3$. 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 $n\ge 6$.

13 pages, 1 figure. v2: final version, to appear in International Mathematics Research Notices