The Prym-canonical Clifford index

The theme of the thesis is Brill--Noether theory for algebraic curves, with a special focus on Prym-canonical curves. In Chapter $1$, given a smooth irreducible Prym curve $(C, \eta)$ of genus $g$, we define two new invariants. The first one is the Prym-canonical Clifford index, $\Cliff_{\eta}(C)$, which differs from the classical Clifford index in that it is computed with respect to the Prym-canonical bundle $\omega_C \otimes \eta$. The second one is the Prym-canonical Clifford dimension of $(C, \eta)$. We prove the Prym-Clifford's Theorem, asserting that $\Cliff_{\eta}(C)\ge 0$, and equality holds if and only if $|\omega_C \otimes \eta|$ has base points. We conclude the chapter with some remarks about the relation between the Prym-canonical Clifford index of a curve and its étale double cover. In Chapter $2$ we classify curves such that $1 \le \Cliff_{\eta}(C)\le 2$. We prove that the first equality occurs if and only if the Prym-canonical bundle is base-point free but not very ample. On the other hand, Prym curves with $\Cliff_{\eta}(C)=2$ are such that the Prym-canonical line bundle $\omega_C \otimes \eta$ embeds $C$ in $\P^{g-2}$ with a trisecant line. Moreover, we compute the Prym-canonical Clifford dimension of bielliptic curves. We analyze hyperelliptic and general Prym curves in Chapter $3$. Both of them have Prym-canonical Clifford dimension $(0,0)$. It turns out that the Prym-canonical Clifford index of a hyperelliptic Prym curve reflects the geometry of the curve, as it depends on the nontrivial $2$-torsion line bundle $\eta$. For particular choices of $\eta$, it equals the maximal possible value $\lfloor \frac{g-1}{2} \rfloor$. By upper semicontinuity, we conclude that the Prym-canonical Clifford index of a general Prym curve $(C, \eta)$ coincides with the classical Clifford index of $C$. Finally, in Chapter $4$ we define Prym-exceptional curves, and with the aim of constructing an example, we consider Prym-canonical curves on Nikulin surfaces. Our approach does not yield the desired result, but we provide a new example of exceptional curves with respect to the Clifford index.