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Very stable and wobbly loci for elliptic curves

Kuntal Banerjee, Steven Rayan

Abstract

We explore very stable and wobbly bundles, twisted in a particular sense by a line bundle, over complex algebraic curves of genus $1$. We verify that twisted stable bundles on an elliptic curve are not very stable for any positive twist. We utilize semistability of trivially twisted very stable bundles to prove that the wobbly locus is always a divisor in the moduli space of semistable bundles on a genus $1$ curve. We prove, by extension, a conjecture regarding the closedness and dimension of the wobbly locus in this setting. This conjecture was originally formulated by Drinfeld in higher genus.

Very stable and wobbly loci for elliptic curves

Abstract

We explore very stable and wobbly bundles, twisted in a particular sense by a line bundle, over complex algebraic curves of genus . We verify that twisted stable bundles on an elliptic curve are not very stable for any positive twist. We utilize semistability of trivially twisted very stable bundles to prove that the wobbly locus is always a divisor in the moduli space of semistable bundles on a genus curve. We prove, by extension, a conjecture regarding the closedness and dimension of the wobbly locus in this setting. This conjecture was originally formulated by Drinfeld in higher genus.

Paper Structure

This paper contains 10 sections, 13 theorems, 56 equations.

Table of Contents

  1. Statement of results
  2. Very stable and wobbly bundles
  3. Twisted global nilpotent cone on $\mathbb{P}^1$
  4. Stable versus very stable bundles
  5. Additivity and multiplicativity of vector bundles on an elliptic curve
  6. Very stable and wobbly loci
  7. Characterization of very stable and wobbly bundles
  8. Topology of the canonically very stable locus and the wobbly locus on curves
  9. Topology of the wobbly locus on an elliptic curve
  10. Computation of topological invariants of the wobbly locus

Key Result

Theorem 1

Let $X$ be a complex elliptic curve, let $r$ be an integer greater than or equal to $2$, let $d$ be any integer, and let $\mathcal{E}(r, d)$ denote the set of isomorphism classes of indecomposable bundles of rank $r$ and degree $d$ on $X$.

Theorems & Definitions (29)

  • Definition 1.0.1
  • Theorem 1
  • Proposition 2.1.1
  • proof
  • Lemma 2.2.1
  • proof
  • Lemma 2.2.2
  • proof
  • Remark 3.1.1
  • Remark 3.1.3
  • ...and 19 more