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Genus $2$ pencils on surfaces with $p_g=K^2=1$, envelopes, and conics tangent to plane cubic curves

Fabrizio Catanese, Noah Ruhland

Abstract

We consider $(1,1)$-surfaces, namely, minimal compact complex surfaces $S$ with $p_g (S) =K_S^2=1$: for these the bicanonical map is a covering of degree $4$ of the plane $\mathbb{P}^2$. And we answer a question posed by Meng Chen, whether they can contain a genus 2 pencil (this is the standard reason of failure of birationality of the bicanonical map). Our main theorem says that those which admit a genus 2 pencil form an irreducible subvariety of codimension $3$ in their moduli space $\frak M_{[1,1]}$; moreover, the general such surface admits exactly $12$ such pencils. The real fun is to relate this variety to the geometry of pencils of conics in the plane everywhere tangent to a cubic curve and a line. We investigate the corresponding variety $\mathcal{T}$ of triples and provide explicit equations using the classical theory of envelopes: among others, equations given in terms of the Weierstrass normal form of the cubic.

Genus $2$ pencils on surfaces with $p_g=K^2=1$, envelopes, and conics tangent to plane cubic curves

Abstract

We consider -surfaces, namely, minimal compact complex surfaces with : for these the bicanonical map is a covering of degree of the plane . And we answer a question posed by Meng Chen, whether they can contain a genus 2 pencil (this is the standard reason of failure of birationality of the bicanonical map). Our main theorem says that those which admit a genus 2 pencil form an irreducible subvariety of codimension in their moduli space ; moreover, the general such surface admits exactly such pencils. The real fun is to relate this variety to the geometry of pencils of conics in the plane everywhere tangent to a cubic curve and a line. We investigate the corresponding variety of triples and provide explicit equations using the classical theory of envelopes: among others, equations given in terms of the Weierstrass normal form of the cubic.
Paper Structure (14 sections, 9 theorems, 91 equations)

This paper contains 14 sections, 9 theorems, 91 equations.

Table of Contents

  1. Part I: standard and non standard case of non-birationality of the bicanonical map
  2. Introduction
  3. First steps of the proof of the main Theorem \ref{['main']}: equivalence of the existence of genus $2$ pencils to a question about quadratic pencils of conics everywhere tangent to a line and a cubic.
  4. How to obtain the pencil of conics $Q_{\lambda}$
  5. End of the proof of the main Theorem \ref{['main']}: existence of the special pencil of conics
  6. Part II : The theory of envelopes and explicit equations for special pencils of conics everywhere tangent to a line and a cubic.
  7. Explicit equations of special everywhere tangent pencils of conics starting from a line $X_1$ and three points on it: the pencil determines the cubic curve $C_1$.
  8. Explicit calculations
  9. Quadratic pencils of conics everywhere tangent to a plane cubic which is smooth (or just not of additive type)
  10. Rank $2$ conics tangent to a smooth plane cubic $C_1$
  11. Explicit equations of special quadratic pencils of conics starting from the Weierstrass normal form of a cubic
  12. Determination of the conics $Q_\lambda$
  13. Finding the linear form $l(x)$
  14. The conics $Q_\lambda$

Key Result

Theorem 2.1

(I) Let $\frak M_{[1,1]}$ be the moduli space of complex minimal surfaces $S$ of general type with $p_g(S)=1, K^2_S=1$, which is irreducible of dimension $18$. Then the general surface $S \in \frak M_{[1,1]}$ does not admit a free genus $2$ pencil, that is, a fibration $\Psi : S \rightarrow C$ with These equations exhibit the bicanonical morphism as an iterated double covering of the plane factor

Theorems & Definitions (22)

  • Theorem 2.1
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Remark 3.3
  • Definition 3.4
  • Proposition 6.1
  • proof
  • ...and 12 more