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Characterization of genuine ramification using formal orbifolds

Indranil Biswas, Manish Kumar, A. J. Parameswaran

Abstract

We give a characterization of genuinely ramified maps of formal orbifolds in the Tannakian framework. In particular we show that a morphism is genuinely ramified if and only if the pullback of every stable bundle remains stable in the orbifold category. We also give some other characterizations of genuine ramification. This generalizes the results of [BKP1] and [BP1]. In fact, it is a positive characteristic analogue of results in [BKP2].

Characterization of genuine ramification using formal orbifolds

Abstract

We give a characterization of genuinely ramified maps of formal orbifolds in the Tannakian framework. In particular we show that a morphism is genuinely ramified if and only if the pullback of every stable bundle remains stable in the orbifold category. We also give some other characterizations of genuine ramification. This generalizes the results of [BKP1] and [BP1]. In fact, it is a positive characteristic analogue of results in [BKP2].
Paper Structure (5 sections, 13 theorems, 48 equations)

This paper contains 5 sections, 13 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Vector bundles over orbifolds
  3. Pullback of stable bundles
  4. Tannakian characterization
  5. Pushforward of the structure sheaf

Key Result

Proposition 1.1

Let $f\,:\,Y\, \longrightarrow \,X$ be a morphism of irreducible smooth curves over an algebraically closed field of characteristic zero. Let $X$ be equipped with a branch data $P$. Then $f_*{\mathcal{O}}_Y$ is a semistable "parabolic bundle" of degree zero belonging to $\mathop{\mathrm{Vect}}\limit

Theorems & Definitions (26)

  • Proposition 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Definition 3.1
  • ...and 16 more