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摘要翻译：
本文通过确定Fermat型曲线的亏格，开始研究加权射影平面上具有一个平凡的权值${mathbb P}（1,m,n)$上的曲线。这些曲线是由一个“齐次”多项式定义的，类似于费马大定理中的多项式。我们首先为平面的标准仿射覆盖寻找局部坐标，然后证明曲线是光滑的。这是通过将曲线向上拉到表面的去模糊化来完成的。然后构造一个从曲线到${\mathbb p^1}$的映射，并确定它的分枝因子。我们的结论是将Hurwitz定理应用于这个映射，得到$C$'s亏格。
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英文标题：
《The genus of a curve of Fermat type》
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作者：
Jeremiah M. Kermes
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最新提交年份：
2007
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分类信息：
一级分类：Mathematics 数学
二级分类：Algebraic Geometry 代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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英文摘要：
In this paper we begin to study curves on a weighted projective plane with one trivial weight, ${\mathbb P}(1,m,n)$, by determining the genus of curves of Fermat type. These are curves defined by a ``homogeneous'' polynomial analagous to the one from Fermat's last theorem. We begin by finding local coordinates for the standard affine cover of the plane, and then prove that the curve is smooth. This is done by pulling the curve up to the surface's desingularization. Then a map from the curve to ${\mathbb P^1}$ is constructed, and it's ramification divisor is determined. We conclude by applying Hurwitz's theorem to this map to obtain $C$'s genus.
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PDF链接：
https://arxiv.org/pdf/0710.3890
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