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Algebraic hyperbolicity for surfaces in smooth projective toric threefolds with Picard rank 2 and 3

Sharon Robins

Abstract

Algebraic hyperbolicity serves as a bridge between differential geometry and algebraic geometry. Generally, it is difficult to show that a given projective variety is algebraically hyperbolic. However, it was established recently that a very general surface of degree at least five in projective space is algebraically hyperbolic. We are interested in generalizing the study of surfaces in projective space to surfaces in smooth projective toric threefolds with Picard rank 2 or 3. Following Kleinschmidt and Batyrev, we explore the combinatorial description of smooth projective toric threefolds with Picard rank 2 and 3. We then use Haase and Ilten's method of finding algebraically hyperbolic surfaces in toric threefolds. As a result, we determine many algebraically hyperbolic surfaces in each of these varieties.

Algebraic hyperbolicity for surfaces in smooth projective toric threefolds with Picard rank 2 and 3

Abstract

Algebraic hyperbolicity serves as a bridge between differential geometry and algebraic geometry. Generally, it is difficult to show that a given projective variety is algebraically hyperbolic. However, it was established recently that a very general surface of degree at least five in projective space is algebraically hyperbolic. We are interested in generalizing the study of surfaces in projective space to surfaces in smooth projective toric threefolds with Picard rank 2 or 3. Following Kleinschmidt and Batyrev, we explore the combinatorial description of smooth projective toric threefolds with Picard rank 2 and 3. We then use Haase and Ilten's method of finding algebraically hyperbolic surfaces in toric threefolds. As a result, we determine many algebraically hyperbolic surfaces in each of these varieties.

Paper Structure

This paper contains 10 sections, 17 theorems, 48 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Approach and results
  4. Organization
  5. Preliminaries
  6. Toric varieties
  7. Algebraic hyperbolicity
  8. Classification of toric varieties with Picard rank 2 or 3
  9. Main results on algebraic hyperbolicity
  10. Proof of main results

Key Result

Theorem 2.1

Let $X_{\Sigma}$ be a smooth complete toric threefold of at most Picard rank 3 and let $E,E'$ be two nef divisor on $X_\Sigma$. Then $(E,E')$ has the IDP.

Figures (4)

  • Figure 1: Nef and effective cone of $X_\Sigma\cong \mathbb{P}(\mathcal{O}_{\mathbb{P}^2}\oplus \mathcal{O}_{\mathbb{P}^2}(l))$
  • Figure 2: Algebraic hyperbolicity for a very general surface of the type $aD_2+bD_3$ in $X_\Sigma\cong \mathbb{P}(\mathcal{O}_{\mathbb{P}^2}\oplus \mathcal{O}_{\mathbb{P}^2}(l))$, for $l=0$ and 2.
  • Figure 3: Algebraic hyperbolicity for a very general surface of the type $aD_3+bD_4$ in $X_{\Sigma} \cong \mathbb{P}(\mathcal{O}_{\mathbb{P}^1}\oplus \mathcal{O}_{\mathbb{P}^1}(l_1)\oplus \mathcal{O}_{\mathbb{P}^1}(l_2))$
  • Figure 4: Facets of Polytope $P(aD_2+bD_3)$ when $a,b\geq1$ in Lemma \ref{['lemma7.1']}

Theorems & Definitions (32)

  • Theorem 2.1: MR2551605,robins2021integer
  • Lemma 2.2: MR2810322
  • Definition 2.3
  • Proposition 2.4: 1903.02681
  • Theorem 2.5: 1903.02681
  • Lemma 2.6: 1903.02681
  • Remark 2.7
  • Corollary 2.8
  • proof
  • Remark 2.9
  • ...and 22 more