Gaussian maps for singular curves on Enriques surfaces

IF 0.7 2区 数学 Q2 MATHEMATICS
Dario Faro
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引用次数: 0

Abstract

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 (SH) be a polarized Enriques surface and let (Cf) 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.

Abstract Image

恩里克曲面上奇异曲线的高斯映射
有标记的普赖姆曲线是一个三元组(((C,\alpha ,T_d)\),其中 C 是一条光滑的代数曲线,\(\alpha \)是 C 上的\(2-\)扭转线束,\(T_d\)是阶数为 d 的除数。我们用高斯映射给出了有标记的普赖姆曲线 \((C,\alpha ,T_d)\)在恩里克斯曲面上的奇点模型的障碍,该奇点模型只有一个乘数为 d 的普通奇点,这样 \(T_d\)就是奇点在归一化映射作用下的回拉。更确切地说,让(S, H)是一个极化的恩里克斯曲面,让(C, f)是一条光滑曲线,同时有一个态(f:C \rightarrow S\ )双向到它的像上,并且使得 \(f(C) \in |H|\),f(C) 恰好有一个乘数为 d 的普通奇异点。让 \(\alpha =f^*\omega _S\) 和 \(T_d\) 是 f(C) 奇点上的除数。我们证明,如果 H 是足够正的,那么 C 上与\(\omega _C\)、\(\alpha \)和\(T_d\)相关的某些自然高斯映射就不是投射性的。相反,我们证明了对于有标记的普赖姆曲线的模空间中的一般三元组 \((C,\alpha ,T_d)\),同样的高斯映射都是可射的。
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来源期刊
Collectanea Mathematica
Collectanea Mathematica 数学-数学
CiteScore
2.70
自引率
9.10%
发文量
36
审稿时长
>12 weeks
期刊介绍: Collectanea Mathematica publishes original research peer reviewed papers of high quality in all fields of pure and applied mathematics. It is an international journal of the University of Barcelona and the oldest mathematical journal in Spain. It was founded in 1948 by José M. Orts. Previously self-published by the Institut de Matemàtica (IMUB) of the Universitat de Barcelona, as of 2011 it is published by Springer.
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