Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Classification of low degree del Pezzo orbifolds

Saptarshi Dandapat

TL;DR

The paper develops a complete classification of low-degree del Pezzo orbifolds with irreducible boundary by degree $d\le5$, placing strong emphasis on the interaction between boundary curves and the positivity of $-(K_{\underline X}+\varepsilon D)$. It systematically lists irreducible boundary curves of prescribed anticanonical degree, and determines, via adjunction, Hodge Index, and Kawamata–Viehweg vanishing, when the orbifold pair is ample, nef, or nef but not big for varied $\varepsilon$ and curve types. This leads to explicit descriptions of the possible boundary configurations across degrees $1$ through $5$ and provides a foundation for constructing free Campana curves on these surfaces. The results also yield a framework for identifying weak del Pezzo orbifolds in this irreducible-boundary setting, with detailed birational realizations of curve classes and their positivity regimes. Overall, the work lays groundwork for further arithmetic and geometric investigations of Campana sections on low-degree del Pezzo orbifolds.

Abstract

In this paper we classify low degree del Pezzo orbifolds with irreducible boundaries. In order to achieve desired boundaries, we classify low degree curves on low degree del Pezzo surfaces. The notion of Campana orbifolds was introduced by Campana in 2004. A del Pezzo orbifold is a Campana orbifold whose underlying surface is a del Pezzo surface. The classification is elementary applications of adjunction formula, Riemann-Roch theorem, Hodge Index theorem and Kawamata-Viehweg vanishing theorem.

Classification of low degree del Pezzo orbifolds

TL;DR

The paper develops a complete classification of low-degree del Pezzo orbifolds with irreducible boundary by degree , placing strong emphasis on the interaction between boundary curves and the positivity of . It systematically lists irreducible boundary curves of prescribed anticanonical degree, and determines, via adjunction, Hodge Index, and Kawamata–Viehweg vanishing, when the orbifold pair is ample, nef, or nef but not big for varied and curve types. This leads to explicit descriptions of the possible boundary configurations across degrees through and provides a foundation for constructing free Campana curves on these surfaces. The results also yield a framework for identifying weak del Pezzo orbifolds in this irreducible-boundary setting, with detailed birational realizations of curve classes and their positivity regimes. Overall, the work lays groundwork for further arithmetic and geometric investigations of Campana sections on low-degree del Pezzo orbifolds.

Abstract

In this paper we classify low degree del Pezzo orbifolds with irreducible boundaries. In order to achieve desired boundaries, we classify low degree curves on low degree del Pezzo surfaces. The notion of Campana orbifolds was introduced by Campana in 2004. A del Pezzo orbifold is a Campana orbifold whose underlying surface is a del Pezzo surface. The classification is elementary applications of adjunction formula, Riemann-Roch theorem, Hodge Index theorem and Kawamata-Viehweg vanishing theorem.
Paper Structure (19 sections, 22 theorems, 10 equations)

This paper contains 19 sections, 22 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Campana orbifolds.
  3. Main results.
  4. Preliminaries
  5. Surfaces of degree 1.
  6. When boundary is irreducible.
  7. Classification of irreducible boundaries.
  8. Surfaces of degree 2.
  9. Low degree curves on $\underline{X}$.
  10. Classification of irreducible boundaries.
  11. Surfaces of degree 3: Cubic surfaces.
  12. Low degree curves on $\underline{X}$.
  13. Classification of irreducible boundaries.
  14. Surfaces of degree 4.
  15. Low degree curves on $\underline{X}$.
  16. ...and 4 more sections

Key Result

Theorem 1.2

Let $(X, \epsilon D)$ is a klt (weak) del Pezzo orbifold such that $\underline{X}$ has degree $d \leq 5$ and $D$ is irreducible $\mathbb{Q}$-divisor of anticanonical degree $-K_{\underline{X}} \cdot D \leq 2d$. Then the pair $(X, \epsilon D)$ must be one of the following options.

Theorems & Definitions (38)

  • Remark 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Proposition 3.2
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • proof
  • Proposition 4.3
  • Proposition 5.1
  • ...and 28 more