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Algebraic hyperbolicity of surfaces in Fano threefolds with Picard number one

Haesong Seo

TL;DR

This work classifies the algebraic hyperbolicity of very general surfaces in general Fano threefolds with Picard number one, establishing sharp degree thresholds relative to the Fano index. Employing a generalized Coskun–Riedl framework, it uses Lazarsfeld–Mukai bundles and section-dominating line bundles (including weighted versions) together with Bott’s theorem on Grassmannians to bound the degrees of curves on surfaces. The authors provide a unified treatment that covers both weighted and non-weighted Fano threefolds, delivering a complete hyperbolicity classification (up to the weighted-hypersurface exception) and refining the hyperbolicity bounds for several deformation classes. The methods yield both a conceptual approach via nefness of certain vector bundles and a practical scroll-argument mechanism to handle challenging degree cases, contributing to the broader hyperbolicity conjecture in higher-dimensional homogeneous settings.

Abstract

In this paper, we study the algebraic hyperbolicity of very general surfaces in general Fano threefolds with Picard number one. We completely classify the algebraically hyperbolicity of those surfaces, except for surfaces in weighted hypersurfaces.

Algebraic hyperbolicity of surfaces in Fano threefolds with Picard number one

TL;DR

This work classifies the algebraic hyperbolicity of very general surfaces in general Fano threefolds with Picard number one, establishing sharp degree thresholds relative to the Fano index. Employing a generalized Coskun–Riedl framework, it uses Lazarsfeld–Mukai bundles and section-dominating line bundles (including weighted versions) together with Bott’s theorem on Grassmannians to bound the degrees of curves on surfaces. The authors provide a unified treatment that covers both weighted and non-weighted Fano threefolds, delivering a complete hyperbolicity classification (up to the weighted-hypersurface exception) and refining the hyperbolicity bounds for several deformation classes. The methods yield both a conceptual approach via nefness of certain vector bundles and a practical scroll-argument mechanism to handle challenging degree cases, contributing to the broader hyperbolicity conjecture in higher-dimensional homogeneous settings.

Abstract

In this paper, we study the algebraic hyperbolicity of very general surfaces in general Fano threefolds with Picard number one. We completely classify the algebraically hyperbolicity of those surfaces, except for surfaces in weighted hypersurfaces.

Paper Structure

This paper contains 10 sections, 9 theorems, 98 equations, 6 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Setup.
  4. Section-dominating family.
  5. Surface scrolls.
  6. Bott's theorem for Grassmannians.
  7. Technical lemmas
  8. Weighted projective space.
  9. Cohomology of homogeneous vector bundles.
  10. Proof of main theorem \ref{['thm:main-theorem']}

Key Result

Theorem 1.2

Let $X$ be a general Fano threefold with Picard number one. Let $\mathcal{O}_X(1)$ be a Picard generator and $r$ the Fano index, i.e., $-K_X = \mathcal{O}_X(r)$ in the Picard group of $X$. For $a \geq 1$, let $S \in |\mathcal{O}_X(a)|$ be a very general surface.

Theorems & Definitions (21)

  • Conjecture 1.1: MY2024
  • Theorem 1.2
  • Proposition 2.1: cf. CR2023
  • proof
  • Definition 2.2: cf. CR2023
  • Example 2.3: cf. CR2023, Mioranci2023
  • Lemma 2.4: cf. CR2023
  • Proposition 2.5: cf. CR2023
  • proof
  • Lemma 2.6: cf. CR2023
  • ...and 11 more