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Toric Elliptic Pairs with Picard Number Three

Aditya Khurmi

Abstract

An elliptic pair $(X, C)$ is a generalization of a rational elliptic fibration $X \to \mathbb{P}^1$ with fiber $C,$ introduced in \cite{jenia_blowup}. Here, $X$ is a projective rational surface with log terminal singularities, and $C$ is an irreducible curve contained in the smooth locus of $X,$ with $p_a(C)=1$ and $C^2=0.$ These naturally arise as blowups $X:=\text{Bl}_e(\mathbb{P}_Δ)$ of projective toric surfaces, whose Newton polygon is elliptic. The order of $\mathcal{O}(C)|_C$ in $\text{Pic}^0(C)$ gives a quantitative way to check if $X$ is an elliptic fibration, which is equivalent to finiteness of the order. We call $Δ$ a Lang-Trotter polygon when this order is infinite, in which case $\overline{\text{Eff}(\text{Bl}_e(\mathbb{P}_Δ))}$ is non-polyhedral. The paper \cite{lizzie} shows there are exactly $3$ elliptic triangles up to $\text{SL}_2(\mathbb{Z}),$ none of which is Lang-Trotter. The paper \cite{jenia_blowup} gives an infinite family of Lang-Trotter pentagons and heptagons, and various examples of other polygons when $ρ(\mathbb{P}_Δ)>2.$ Remark 4.7 in the paper asks if any Lang-Trotter quadrilaterals exist, and we answer this in the negative by studying the curves in the Zariski Decomposition of $K_X+C.$

Toric Elliptic Pairs with Picard Number Three

Abstract

An elliptic pair is a generalization of a rational elliptic fibration with fiber introduced in \cite{jenia_blowup}. Here, is a projective rational surface with log terminal singularities, and is an irreducible curve contained in the smooth locus of with and These naturally arise as blowups of projective toric surfaces, whose Newton polygon is elliptic. The order of in gives a quantitative way to check if is an elliptic fibration, which is equivalent to finiteness of the order. We call a Lang-Trotter polygon when this order is infinite, in which case is non-polyhedral. The paper \cite{lizzie} shows there are exactly elliptic triangles up to none of which is Lang-Trotter. The paper \cite{jenia_blowup} gives an infinite family of Lang-Trotter pentagons and heptagons, and various examples of other polygons when Remark 4.7 in the paper asks if any Lang-Trotter quadrilaterals exist, and we answer this in the negative by studying the curves in the Zariski Decomposition of

Paper Structure

This paper contains 20 sections, 47 theorems, 70 equations, 20 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Elliptic Pairs and Polygons
  3. Semi-Elliptic Quadrilaterals
  4. Case Division
  5. Opposite Du Val case
  6. Adjacent Du Val case
  7. A General Analysis on the Combinatorial Configuration
  8. The $A_n$ case subdivision
  9. The $DE$ cases
  10. Classification
  11. The smooth case
  12. Classification
  13. Case I
  14. Case II
  15. The $A_n$ case
  16. ...and 5 more sections

Key Result

Theorem 1.8

There are no Lang-Trotter quadrilaterals.

Figures (20)

  • Figure 1: Left: The smooth blown up toric surface $Y$ for $\Delta,$ with a potential $(-1)$ curve intersecting two toric boundary curves. The blue curves form the exceptional divisor of $Y \to X.$ Any blue curve whose self-intersection has not been written is a $(-2)$ curve. Right: The dual graph of the relevant locus under the contractions $Y \to \dots \to W.$
  • Figure 2: Left: The toric fan (notation from fulton). Right: (Part of) the corresponding Newton polygon.
  • Figure 3: Contraction introduces a singular point on $R$ if $\widetilde{R}$ passes through torus invariant points in the exceptional locus
  • Figure 4: The tree of cases
  • Figure 5: The (unordered) signature is $(+,+,-,-)$
  • ...and 15 more figures

Theorems & Definitions (113)

  • Definition 1.1
  • Example 1.2
  • Remark 1.3
  • Definition 1.4
  • Remark 1.5
  • Definition 1.6
  • Theorem 1.8
  • Theorem 1.9
  • Corollary 1.10
  • Definition 1.11
  • ...and 103 more