Towards the Bertram-Feinberg-Mukai conjecture

In this paper, we prove the existence portion of the Bertram-Feinberg-Mukai Conjecture for an infinite family of new cases using degeneration technique. This not only leads to a substantial improvement of known results but also develops finer tools for analyzing the moduli of rank two limit linear series which should be useful for applications to other higher-rank Brill-Noether problems.