Rationality of weighted hypersurfaces of special degree

TL;DR

This paper characterizes the rationality of certain weighted hypersurfaces using algebraic and numerical methods, providing new examples of rational varieties and analyzing the rationality of specific affine threefolds.

Contribution

It offers a complete characterization of rationality for weighted hypersurfaces under specific divisibility conditions, combining algebraic proofs with numerical semigroup results.

Findings

Characterization of rationality when L divides the degree

New examples of rational varieties with quotient singularities

Determination of rationality for affine Pham-Brieskorn threefolds

Abstract

Let $X \subset \mathbb{P}(w_0, w_1, w_2, w_3)$ be a quasismooth well-formed weighted projective hypersurface and let $L = lcm(w_0,w_1,w_2,w_3)$. We characterize when $X$ is rational under the assumption that $L$ divides $deg(X)$ by combining an algebraic proof of rationality valid in all dimensions with a new result on numerical semigroups. As applications, we give new examples of families of normal projective rational varieties with quotient singularities and ample canonical divisor; we also determine precisely which affine Pham-Brieskorn threefolds are rational.