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Let $V$ be a vector bundle over a smooth curve $C$. In this paper, we studytwisted Brill--Noether loci parametrising stable bundles $E$ of rank $n$ anddegree $e$ with the property that $h^0 (C, V \otimes E) \ge k$. We prove that,under conditions similar to those of Teixidor i Bigas and of Mercat, theBrill-Noether loci are nonempty, and in many cases have a component which isgenerically smooth and of the expected dimension. Along the way, we prove theirreducibility of certain components of both twisted and "nontwisted"Brill--Noether loci. We describe the tangent cones to the twisted Brill-Noetherloci. We end with an example of a general bundle over a general curve havingpositive-dimensional twisted Brill--Noether loci with negative expecteddimension.

Nonemptiness and smoothness of twisted Brill-Noether loci