Special Divisors on Real Trigonal Curves

In this paper we examine the topology of Brill-Noether varieties associated to real trigonal curves. More precisely, we aim to count the connected components of the real locus of the varieties parametrizing linear systems of degree $d$ and dimension at least $r$. We do this count when the relations $m=g-d+r-1\le d-2r-1$ are satisfied, where $m$ is the Maroni invariant and $g$ is the genus of the curve.