A pathological case of the $C_1$ conjecture in mixed characteristic

Abstract

Let $K$ be a field of characteristic 0. Fix integers $r,d$ coprime with $r\ge 2$. Let $X_K$ be a smooth, projective, geometrically connected curve of genus $g\ge 2$ defined over K. Assume there exists a line bundle $L_K$ on $X_K$ of degree $d$. In this article we prove the existence of a stable locally free sheaf on $X_K$ with rank $r$ and determinant $L_K$. This trivially proves the $C_1$ conjecture in mixed characteristic for the moduli space of stable locally free sheaves of fixed rank and determinant over a smooth, projective curve.

Published in Mathematical Proceedings of the Cambridge Philosophical Society

Cited by in corpus (3)

• Existence of semistable vector bundles with fixed determinants
• Examples of varieties with index one on $C_1$-fields
• Descent theory of simple sheaves on $C_1$-fields