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≥2. Let XR be a regular fibred surface over Spec(R) with special fibre denoted Xk, a generalised tree-like curve of genus g≥2. Let LR be a line bundle on XR of degree d such that the degree of the restriction of LR on the rational components of Xk is a multiple of r. In this article we prove the existence of a rank r locally free sheaf on XR of determinant LR such that it is semistable on the fibres.