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Birational geometry of hypersurfaces in products of weighted projective spaces

Francesco Antonio Denisi

Abstract

We study the birational geometry of hypersurfaces in products of weighted projective spaces, extending results previously established by J. C. Ottem. For most cases where these hypersurfaces are Mori dream spaces, we determine all relevant cones and characterise their birational models, along with the small $\mathbf{Q}$-factorial modifications to them. We also provide a presentation of their Cox ring. Finally, we establish the birational form of the Kawamata-Morrison cone conjecture for terminal Calabi-Yau hypersurfaces in Gorenstein products of weighted projective spaces.

Birational geometry of hypersurfaces in products of weighted projective spaces

Abstract

We study the birational geometry of hypersurfaces in products of weighted projective spaces, extending results previously established by J. C. Ottem. For most cases where these hypersurfaces are Mori dream spaces, we determine all relevant cones and characterise their birational models, along with the small -factorial modifications to them. We also provide a presentation of their Cox ring. Finally, we establish the birational form of the Kawamata-Morrison cone conjecture for terminal Calabi-Yau hypersurfaces in Gorenstein products of weighted projective spaces.

Paper Structure

This paper contains 10 sections, 16 theorems, 66 equations.

Table of Contents

  1. Conventions and notations
  2. Introduction
  3. Preliminaries
  4. Hypersurfaces in products of two weighted projective spaces
  5. Proof of Theorem \ref{['thmA']}
  6. Proof of Theorem \ref{['thmB']}
  7. The birational geometry of $Y$ via toric geometry
  8. Calabi-Yau multi-weighted hypersurfaces and the cone conjecture
  9. Proof of Proposition \ref{['propCY']}
  10. Proof of Theorem \ref{['thmC']}

Key Result

Theorem 2.1

Let $X$ be a product $\mathbf{P}^n(\underline{v}) \times \mathbf{P}^m(\underline{w})$, with $\mathrm{dim}(X)\geq 4$, and $\mathscr{O}_X(d,e)$ a line bundle on $X$, where $d$ and $e$ are positive integers. Furthermore, set $a(\underline{w}) := \mathrm{l.c.m.}\{w_i\}_{i=0}^m$.

Theorems & Definitions (36)

  • Theorem 2.1
  • Theorem 2.2
  • Proposition 2.3
  • Theorem 2.4
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Definition 3.5
  • Definition 3.6
  • ...and 26 more