Rationality of weighted hypersurfaces of special degree

Author:

Michael Chitayat

Keyword:

Mathematics, Algebraic Geometry, Algebraic Geometry (math.AG), Commutative Algebra (math.AC)

journal:

date:

2023-08-09 16:00:00

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.

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