Gaussian maps for singular curves on Enriques surfaces

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2024年4月30日

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Dario Faro

摘要

A marked Prym curve is a triple $(C,\alpha ,T_d)$ $(C, α, T_{d})$ where C is a smooth algebraic curve, $\alpha $ $α$ is a $2-$ $2 -$ torsion line bundle on C, and $T_d$ $T_{d}$ is a divisor of degree d. We give obstructions—in terms of Gaussian maps—for a marked Prym curve $(C,\alpha ,T_d)$ $(C, α, T_{d})$ to admit a singular model lying on an Enriques surface with only one ordinary singular point of multiplicity d, such that $T_d$ $T_{d}$ 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 $f:C \rightarrow S$ $f : C \to S$ birational onto its image and such that $f(C) \in |H|$ $f (C) \in | H |$ , f(C) has exactly one ordinary singular point of multiplicity d. Let $\alpha =f^*\omega _S$ $α = f^{*} ω_{S}$ and $T_d$ $T_{d}$ 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 $\omega _C$ $ω_{C}$ , $\alpha $ $α$ , and $T_d$ $T_{d}$ are not surjective. On the contrary, we show that for the general triple in the moduli space of marked Prym curves $(C,\alpha ,T_d)$ $(C, α, T_{d})$ , the same Gaussian maps are surjective.