Resumen de Gaussian maps for singular curves on Enriques surfaces

Dario Faro

• 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.