  Higher rank Brill-Noether theory for genus 6 is especially interesting as,
even in the general case, some unexpected phenomena arise which are absent in
lower genus. Moreover, it is the first case for which there exist curves of
Clifford dimension greater than 1 (smooth plane quintics). In all cases, we
obtain new upper bounds for non-emptiness of Brill-Noether loci and construct
many examples which approach these upper bounds more closely than those that
are well known. Some of our examples of non-empty Brill-Noether loci have
negative Brill-Noether numbers.
