Existence of semistable vector bundles with fixed determinants

Abstract

Let $R$ be an excellent Henselian discrete valuation ring with algebraically closed residue field $k$ of any characteristic. Fix integers $r,d$ with $r\ge 2$. Let $X_R$ be a regular fibred surface over Spec($R$) with special fibre denoted $X_k$, a generalised tree-like curve of genus $g\ge 2$. Let $L_R$ be a line bundle on $X_R$ of degree $d$ such that the degree of the restriction of $L_R$ on the rational components of $X_k$ is a multiple of $r$. In this article we prove the existence of a rank $r$ locally free sheaf on $X_R$ of determinant $L_R$ such that it is semistable on the fibres.

Published in Journal of Geometry and Physics

References in corpus (2)

• Moduli spaces of SL(r)-bundles on singular irreducible curves
• A pathological case of the $C_1$ conjecture in mixed characteristic

Cited by in corpus (1)

• Néron models of intermediate Jacobians associated to moduli spaces