We compute the class of the closure of the locus of hyperelliptic curves in the moduli space of stable genus-3 curves in terms of the tautological class $lambda$ and the boundary classes $delta_0$ and $delta_1$. The expression of this class is known,
but here we compute it directly, by means of Porteous Formula, without resorting to blowups or test curves.
In this paper we consider the Brill-Noether locus $W_{underline d}(C)$ of line bundles of multidegree $underline d$ of total degree $g-1$ having a nonzero section on a nodal reducible curve $C$ of genus $ggeq2$. We give an explicit description of the
irreducible components of $W_{underline d}(C)$ for a semistable multidegre $underline d$. As a consequence we show that, if two semistable multidegrees of total degre $g-1$ on a curve with no rational components differ by a twister, then the respective Brill-Noether loci have isomorphic components.
