The Prym-Petri expectation for double covers of curves on Nikulin surfaces

Let $(C,\eta)\in \mathcal R_g$ be general, and let

$$\pi:\widetilde C\rightarrow C$$

be the étale double cover defined by $\eta$. Write $W^r_{2g-2}(\widetilde C)$ for the Brill–Noether locus of line bundles of degree $2g-2$ with at least $r+1$ sections, $V^r(C,\eta)$ for the corresponding Prym-Brill–Noether locus, and $\mu_{0,L}$ and $\mu_{0,L}^+$ for the full and invariant Prym-Petri maps.

Prym-Petri expectation. If $-r\leq \rho(2g-1,r,2g-2)<r$, then

$$W^r_{2g-2}(\widetilde C)=V^r(C,\eta).$$

In particular, for all $L\in V^r(C,\eta)$ one has

$$\dim\operatorname{Ker}\mu_{0,L}=\dim\operatorname{Ker}\mu_{0,L}^+=-\rho^+(g,r).$$

If $\rho(2g-1,r,2g-2)\geq r$, then both $\mu_{0,L}$ and $\mu_{0,L}^+$ are injective for all $L\in V^r(C,\eta)$.

This describes the expected behavior of the Prym-Petri maps for a general étale double cover: in the intermediate range the ordinary and Prym Brill–Noether loci coincide locally through the stated kernel-dimension equality, while beyond that range both multiplication maps are expected to be injective. The supplied context presents this as an expectation rather than giving evidence that it has been proved.