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Sections of rational elliptic Lefschetz fibrations

Mohan Bhupal, Sergey Finashin

TL;DR

The paper develops a comprehensive framework for classifying sections (lines) in elliptic Lefschetz fibrations by translating geometric data into monodromy factorizations in pure mapping class groups and their liftings to $\mathrm{PMod}(F_d)$. It proves a homology-based correspondence between isotopy classes of sections and $(-1)$-curves, constructs explicit liftings for del Pezzo surfaces of degrees $d=1,2,3$, and shows a bijection between section isotopy classes and lifted factorization conjugacy classes, with counts $n_d=240,56,27$. A central insight is the presentation of monodromy factorizations through the $E_8$ root lattice, providing a concrete link between factorization data and root vectors, including a full description for $d=1$. The results illuminate the role of Hurwitz moves in relating factorizations and demonstrate a universal structure: all liftings of a fixed base factorization correspond to the same geometric classes across deformation-equivalent settings, offering a robust toolbox for understanding sections in rational elliptic surfaces and their del Pezzo degenerations.

Abstract

We give a list of monodromy factorizations in the pure mapping class group $Mod(T_{d+1})$ of a torus with d+1 marked points that represent lines on a del Pezzo surface Y of degree $d\le4$. These factorizations are lifts of a certain fixed monodromy factorization in $Mod(T_d)$ that represents Y. In the case d=1, discussed in more detail, we give an explicit correspondence between such factorizations and the 240 roots of $E_8=K^\perp$, the orthogonal complement in $H_2(Y)$ of the canonical class.

Sections of rational elliptic Lefschetz fibrations

TL;DR

The paper develops a comprehensive framework for classifying sections (lines) in elliptic Lefschetz fibrations by translating geometric data into monodromy factorizations in pure mapping class groups and their liftings to . It proves a homology-based correspondence between isotopy classes of sections and -curves, constructs explicit liftings for del Pezzo surfaces of degrees , and shows a bijection between section isotopy classes and lifted factorization conjugacy classes, with counts . A central insight is the presentation of monodromy factorizations through the root lattice, providing a concrete link between factorization data and root vectors, including a full description for . The results illuminate the role of Hurwitz moves in relating factorizations and demonstrate a universal structure: all liftings of a fixed base factorization correspond to the same geometric classes across deformation-equivalent settings, offering a robust toolbox for understanding sections in rational elliptic surfaces and their del Pezzo degenerations.

Abstract

We give a list of monodromy factorizations in the pure mapping class group of a torus with d+1 marked points that represent lines on a del Pezzo surface Y of degree . These factorizations are lifts of a certain fixed monodromy factorization in that represents Y. In the case d=1, discussed in more detail, we give an explicit correspondence between such factorizations and the 240 roots of , the orthogonal complement in of the canonical class.

Paper Structure

This paper contains 41 sections, 30 theorems, 13 equations, 5 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Motivations
  3. Homology characterization and isotopy classification theorems
  4. Conventions and terminology
  5. Acknowledgements
  6. Preliminaries
  7. Monodromy factorization of Lefschetz fibrations
  8. LF with sections
  9. Lefschetz thimbles and matching cycles
  10. Elliptic Lefschetz fibrations
  11. Del Pezzo surfaces
  12. Homology classes of sections
  13. Intersection of sections with matching cycles
  14. Monodromy factorizations of $E(1)$ with up to four sections
  15. Sample factorizations and their variations
  16. ...and 26 more sections

Key Result

Theorem 1.1.1

Any pair of rational elliptic LFs $f_j\colon X_j\to S^2$, $j=1,2$, endowed with $1\le m\le7$, or $m=9$ disjoint sections $S_1^j,\dots,S_m^j\subset X_j$ are related by a diffeomorphism $\Phi\colon X_1\to X_2$, such that $f_2\circ\Phi=\phi\circ f_1$ for some diffeomorphism $\phi\colon S^2\to S^2$ and

Figures (5)

  • Figure 1: A system of arcs in $S^2$ representing the monodromy factorization $t_{c_{12}}\cdots t_{c_1}$.
  • Figure 2: Intersection index of a section $S_2$ with a matching cycle $\mathcal{V}_p$. Region $A\subset\mathbb T\smallsetminus\{q_1\}$ with $\partial A=\partial D_1-\partial D_0$ is shaded.
  • Figure 3: Basic torus fibre with marked points $q_1$ (corners) and $q_2$ (center)
  • Figure 4: Intersection graphs $E_8,E_7,E_6$ formed by matching cycles.
  • Figure 5: Action of half-twists $\sigma_p$ on the monodromy factorizations: (a) block-variation, and (b) two substitutions.

Theorems & Definitions (50)

  • Theorem 1.1.1
  • Theorem 1.1.2
  • Theorem 1.2.1
  • Theorem 1.2.2
  • Theorem 1.2.3
  • Theorem 1.2.4
  • Theorem 2.2.1
  • proof
  • Theorem 2.4.1
  • Proposition 2.6.1
  • ...and 40 more