Gaussian maps for singular curves on Enriques surfaces

A marked Prym curve is a triple \((C,\alpha ,T_d)\) where C is a smooth algebraic curve, \(\alpha \) is a \(2-\)torsion line bundle on C, and \(T_d\) is a divisor of degree d. We give obstructions—in terms of Gaussian maps—for a marked Prym curve \((C,\alpha ,T_d)\) to admit a singular model lying on an Enriques surface with only one ordinary singular point of multiplicity d, such that \(T_d\) is the pull-back of the singular point by the normalization map. More precisely, let (S, H) be a polarized Enriques surface and let (C, f) be a smooth curve together with a morphism \(f:C \rightarrow S\) birational onto its image and such that \(f(C) \in |H|\), f(C) has exactly one ordinary singular point of multiplicity d. Let \(\alpha =f^*\omega _S\) and \(T_d\) be the divisor over the singular point of f(C). We show that if H is sufficiently positive then certain natural Gaussian maps on C, associated with \(\omega _C\), \(\alpha \), and \(T_d\) are not surjective. On the contrary, we show that for the general triple in the moduli space of marked Prym curves \((C,\alpha ,T_d)\), the same Gaussian maps are surjective.