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Constructing Lefschetz Fibrations with Arbitrary Slope

Tulin Altunoz, Adalet Cengel

TL;DR

The paper proves that slopes of genus‑g Lefschetz fibrations over S^2 can realize any value in (2,8) by constructing large families with slopes densely filling the interval via fiber sums and mapping‑class‑group substitutions (Matsumoto/ generalized star for high slopes; Lowslope methods for low slopes). It provides explicit mechanisms to push the slope toward 8 and toward 2, yielding a rich spectrum of examples, and shows the maximum attainable slope is 8 in the limit. A key contribution is a counterexample demonstrating that a fibration with only nonseparating vanishing cycles and slope 4−4/g need not be hyperelliptic, clarifying the relationship between slope and hyperellipticity. Together, these results deepen the understanding of the geography of Lefschetz fibrations and the role of monodromy substitutions in shaping topological invariants.

Abstract

We prove that for any real number $r\in (2,8)$, there exists a genus-$g$ Lefschetz fibration over the two-sphere with large enough genus-$g$ having the slope arbitrarily close to $r$. It is known that any genus-$g$ hyperelliptic Lefschetz fibration with only nonseparating vanishing cycles has the slope $4-4/g$. We prove that the converse does not hold by showing that there exists a nonhyperelliptic genus-$g$ Lefschetz fibration with only nonseparating vanishing cycles having slope $4-4/g$.

Constructing Lefschetz Fibrations with Arbitrary Slope

TL;DR

The paper proves that slopes of genus‑g Lefschetz fibrations over S^2 can realize any value in (2,8) by constructing large families with slopes densely filling the interval via fiber sums and mapping‑class‑group substitutions (Matsumoto/ generalized star for high slopes; Lowslope methods for low slopes). It provides explicit mechanisms to push the slope toward 8 and toward 2, yielding a rich spectrum of examples, and shows the maximum attainable slope is 8 in the limit. A key contribution is a counterexample demonstrating that a fibration with only nonseparating vanishing cycles and slope 4−4/g need not be hyperelliptic, clarifying the relationship between slope and hyperellipticity. Together, these results deepen the understanding of the geography of Lefschetz fibrations and the role of monodromy substitutions in shaping topological invariants.

Abstract

We prove that for any real number , there exists a genus- Lefschetz fibration over the two-sphere with large enough genus- having the slope arbitrarily close to . It is known that any genus- hyperelliptic Lefschetz fibration with only nonseparating vanishing cycles has the slope . We prove that the converse does not hold by showing that there exists a nonhyperelliptic genus- Lefschetz fibration with only nonseparating vanishing cycles having slope .

Paper Structure

This paper contains 7 sections, 6 theorems, 40 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lefschetz fibrations
  4. Lefschetz fibrations with arbitrary slope
  5. Low--slope Lefschetz fibrations
  6. High--slope Lefschetz fibrations
  7. A counterexample

Key Result

Theorem 1.1

LowslopeLefschetz. Assume $g \geq 2$. For each genus-$g$ Lefschetz fibration, there exists another genus-$g$ Lefschetz fibration with slope less(greater respectively) than $\lambda_f$. Moreover there is no genus-$g$ Lefschetz fibration whose slope is equal to the infimum(supremum respectively) of sl

Figures (2)

  • Figure 3: The curves in generalized star relation $S_h$ on $\Sigma_g$.
  • Figure 4: The curves in the hyperelliptic relator $h_g$ and the curve $d_2$ on $\Sigma_g$.

Theorems & Definitions (11)

  • Theorem 1.1
  • Conjecture 1.2
  • Proposition 2.1
  • Lemma 2.2
  • Remark 3.1
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • proof
  • Theorem 4.1
  • ...and 1 more