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Hyperelliptically fibred surfaces with nodes

Edoardo Ballico, Elizabeth Gasparim, Bruno Suzuki

Abstract

Using elementary methods of algebraic geometry, we present constructions of hyperelliptically fibred surfaces containing nodal fibres.

Hyperelliptically fibred surfaces with nodes

Abstract

Using elementary methods of algebraic geometry, we present constructions of hyperelliptically fibred surfaces containing nodal fibres.
Paper Structure (16 sections, 10 theorems, 25 equations, 9 figures)

This paper contains 16 sections, 10 theorems, 25 equations, 9 figures.

Table of Contents

  1. Motivation
  2. Fibrations containing singular fibres
  3. Elliptic fibrations and elliptic surfaces
  4. Case $S=\emptyset$
  5. Case $S\ne \emptyset$
  6. Fibrations by curves of genus $2$
  7. $c_2(X)$ and $c_1(X)^2$
  8. Examples of hyperelliptic fibrations with nodes
  9. Examples of genus 2 fibrations with nodes
  10. Case $S\ne \emptyset$
  11. Families of hyperelliptic curves in $\mathbb{P}^1\times \mathbb{P}^1$
  12. Fibrations with smooth fibres
  13. Smooth semistable fibrations
  14. Fibrations with general fibre hyperelliptic
  15. Kodaira fibrations
  16. ...and 1 more sections

Key Result

Theorem 2.2

bhpv Let $f\colon X\to Y$ be a smooth holomorphic family of compact complex manifolds. The holomorphic map $f$ is locally trivial (in the Euclidean topology) over $Y$ if and only if all fibres of $f$ are biholomorphic.

Figures (9)

  • Figure 1: Types of nodes on genus 2 curves
  • Figure 2: Family without singular fibres
  • Figure 3: Elliptic family with 1 node
  • Figure 4: Hyperelliptic family with irreducible singular fibre
  • Figure 5: Family with 1 irreducible fibre having 2 nodes
  • ...and 4 more figures

Theorems & Definitions (26)

  • Remark 2.1
  • Theorem 2.2
  • Remark 2.3
  • Theorem 2.4
  • Remark 2.5
  • Definition 2.6
  • Theorem 2.7
  • Theorem 2.8
  • Corollary 2.9
  • Theorem 2.10
  • ...and 16 more