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Real line subbundles of real bundles on curves

Daniel A. Santiago Alvarez

Abstract

For a stable real bundle $E$ of rank $2$ and degree $1$ on a real genus $2$ curve, we describe the action of the real structure of the curve on the set of $4$ maximal line subbundles of degree $0$ of $E$. This describes the Galois action on the set of lines through a real point in the moduli space of such bundles, and is a real algebraic extension of classical work of Newstead. Our proof is an application of techniques of Atiyah from the 1950's. We prove also results on real line subbundles in higher genus using work of Lange-Narasimhan.

Real line subbundles of real bundles on curves

Abstract

For a stable real bundle of rank and degree on a real genus curve, we describe the action of the real structure of the curve on the set of maximal line subbundles of degree of . This describes the Galois action on the set of lines through a real point in the moduli space of such bundles, and is a real algebraic extension of classical work of Newstead. Our proof is an application of techniques of Atiyah from the 1950's. We prove also results on real line subbundles in higher genus using work of Lange-Narasimhan.
Paper Structure (14 sections, 23 theorems, 36 equations, 4 figures)

This paper contains 14 sections, 23 theorems, 36 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Outline of paper
  3. Acknowledgements
  4. Real curves and real hyperelliptic curves
  5. Real hyperelliptic curves
  6. Genus 2
  7. Real holomorphic line bundles
  8. Real Extension lines
  9. Real Extension lines and subbundles in genus $2$
  10. Atiyah's work
  11. Real bundles on real hyperelliptic curves of genus $2$.
  12. Applications of work of Lange-Narasimhan
  13. Topological types in genus $2$
  14. Higher genus results

Key Result

Theorem 1.1

Let $(\Sigma,\tau)$ be a real curve of genus $2$ with nonempty set of real points $\Sigma^{\tau}$. Depending on the topological type of $\tau$ and $\Lambda \in \mathrm{Pic}^1(\Sigma)^{\tau}$, exactly one of the following three behaviors occurs

Figures (4)

  • Figure 1: A real hyperelliptic genus $2$ curve of type $(n,a) = (1,0)$ with one fixed circle and no real Weierstrass points.
  • Figure 2: A real hyperelliptic genus $2$ curve of type $(3,0)$ with three fixed circles and six real Weierstrass points.
  • Figure 3: From left to right, real genus $2$ curves of types $(1,1),(2,1),(0,1)$. These examples have $1,2,0$ fixed circles respectively, and $2,4,0$ real Weierstrass points. Only the first two are real hyperelliptic.
  • Figure 4: The family used in the proof of Theorem \ref{['Non-Surjective-Types-Theorem']}. One chooses $A$ in the fixed circle $C_1$, and $B,C$ in the arc $C_2$ consisting of points on which $\tau(p) = \iota(p)$. By taking $B = \tau(C)$, we obtain a $2$ dimensional subfamily of real divisors.

Theorems & Definitions (38)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Proposition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Lemma 3.1
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • ...and 28 more