We establish, for smooth projective real curves, an analogue of the classical Clifford inequality known for complex curves. We also study the cases when equality holds.

Let ${ℳ}_g$ be the moduli space of smooth complex projective curves of genus $g$. Here we prove that the subset of ${ℳ}_g$ formed by all curves for which some Brill-Noether locus has dimension larger than the expected one has codimension at least two in ${ℳ}_g$. As an application we show that if $X\in {ℳ}_g$ is defined over $ℝ$, then there exists a low degree pencil $uX\to {\mathbb{P}}^1$ defined over $ℝ$.

Here we study the Brill-Noether theory of “extremal” Cornalba’s theta-characteristics on stable curves C of genus g, where “extremal” means that they are line bundles on a quasi-stable model of C with #(Sing(C)) exceptional components

We show that the set of smooth curves of genus $g\ge 0$ admitting a branched covering $X\to {\mathbb{P}}^1$ with only triple ramification points is of dimension at least $max(2g-3,g)$. In characteristic two, such curves have tame rational functions and an analog of Belyi’s Theorem applies to them.