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Rationality of weighted hypersurfaces of special degree

Michael Chitayat

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.

Rationality of weighted hypersurfaces of special degree

Abstract

Let be a quasismooth well-formed weighted projective hypersurface and let . We characterize when is rational under the assumption that divides 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.
Paper Structure (5 sections, 25 theorems, 52 equations)

This paper contains 5 sections, 25 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. A Semigroup Theorem
  3. Rationality of Some Weighted Hypersurfaces
  4. The Rational Affine Pham-Brieskorn Threefolds
  5. Proof of Theorem A

Key Result

Proposition 1

Let $a,c \in \mathbb{N}^+$ and consider the graded polynomial ring $R = {\rm \bf k}_{c,\dots, c, a, \dots, a}[X_1, \dots, X_k, Y_1, \dots, Y_\ell]$ where ${\rm \bf k}$ is a field, $k \geq 1$, $\ell \geq 1$, $\deg(X_i) = c$ and $\deg(Y_j) = a$ for all $i,j$. Suppose $f = g(X_1, \dots, X_k) + h(Y_1, \

Theorems & Definitions (57)

  • Proposition
  • Theorem
  • Theorem
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 3.4
  • proof
  • Proposition 3.5
  • proof
  • ...and 47 more