For $r\geq 3$ and $g= \frac {r(r+1)}{2}$, we study the Prym-Brill-Noether variety $V^r(C,\eta )$ associated to Prym curves $[C,\eta ]$. The locus $\mathcal {R}_g^r$ in $\mathcal {R}_g$ parametrizing Prym curves $(C, \eta )$ with nonempty $V^r(C,\eta )$ is a divisor. We compute some key coefficients of the class $[\overline {\mathcal {R}}_g^r]$ in $\mathrm {Pic}_{\mathbb {Q}}(\overline {\mathcal {R}}_g)$. Furthermore, we examine a strongly Brill-Noether divisor in $\overline {\mathcal {M}}_{g-1,2}$: we show its irreducibility and compute some of its coefficients in $\mathrm {Pic}_{\mathbb {Q}}(\overline {\mathcal {M}}_{g-1,2})$. As a consequence of our results, the moduli space $\mathcal {R}_{14,2}$ is of general type.

We give a large family of weighted projective planes, blown up at a smooth point, that do not have finitely generated Cox rings. We then use the method of Castravet and Tevelev to prove that the moduli space $\overline{M}_{0,n}$ of stable $n$-pointed genus-zero curves does not have a finitely generated Cox ring if $n$ is at least $13$.

We prove the unirationality of the moduli space of complex curves of genus 14. The method essentially relies on linkage of curves. In particular it is shown that a general curve of genus 14 admits a projective model D, in a six-dimensional projective space, which is linked to a general curve C of degree 14 and genus 8 by a complete intersection of quadrics. Using this property we are able to obtain, after a reasonable amount of further work, the unirationality result in the case of genus 14. Moreover, some variations of the same method, involving the Hilbert schemes of curves of very low genus, are used to obtain the same result for the known cases of genus 11, 12, 13.

We study several types of curves and higher-dimensional objects inside the moduli spaces of curves, insisting on their arithmetic properties in the perspective of Grothendieck–Teichmüller theory. On the way we explicitly identify those curves which were originally associated by W. Veech with certain rational polygonal billiards.

AMS 2000 Mathematics subject classification: Primary 14D22; 32G15; 14H30