Mordell-Weil groups of elliptic threefolds and the Alexander module of plane curves
Mordell-Weil groups of elliptic threefolds and the Alexander module of plane curves
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椭圆三重的 Mordell-Weil 群和平面曲线的 Alexander 模
DOI：
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发表时间：
2010
期刊：
影响因子：
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通讯作者：
A. Libgober
中科院分区：
文献类型：
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作者：
J. Cogolludo;A. Libgober
我们证明，以节点和尖点为唯一奇点的不可约平面代数曲线的亚历山大多项式的次数不超过 ${5 \over 3}d-2$，其中 $d$ 是曲线的次数。我们还证明，奇点为节点和尖点的不可约曲线 $C=\{F=0\}\subset \mathbb P^2$ 的亚历山大多项式 $\Delta_C(t)$ 是非平凡的，当且仅当存在齐次多项式 $f$、$g$ 和 $h$ 使得 $f^3+g^2+Fh^6=0$ 时。这是由于此处描述的亚历山大多项式与函数域上某些三重的 Mordell-Weil 群的等级之间的对应关系而获得的。所有结果也都扩展到可约曲线和亚历山大多项式 $\Delta_{C,\epsilon}(t)$ 对应于满射 $\epsilon: \pi_1(\mathbb P^2\setminus C_0 \cup C) \rightarrow \mathbb Z$ 的情况，其中 $C_0$ 是无穷远直线。另外，我们用$\Delta_{C,\epsilon}(t)$根的重数来详细描述上述$F$的关系集合。这种概括是在更大类奇点的背景下进行的，即导致椭圆型有理轨道折叠的奇点。
We show that the degree of the Alexander polynomial of an irreducible plane algebraic curve with nodes and cusps as the only singularities does not exceed ${5 \over 3}d-2$ where $d$ is the degree of the curve. We also show that the Alexander polynomial $\Delta_C(t)$ of an irreducible curve $C=\{F=0\}\subset \mathbb P^2$ whose singularities are nodes and cusps is non-trivial if and only if there exist homogeneous polynomials $f$, $g$, and $h$ such that $f^3+g^2+Fh^6=0$. This is obtained as a consequence of the correspondence, described here, between Alexander polynomials and ranks of Mordell-Weil groups of certain threefolds over function fields. All results also are extended to the case of reducible curves and Alexander polynomials $\Delta_{C,\epsilon}(t)$ corresponding to surjections $\epsilon: \pi_1(\mathbb P^2\setminus C_0 \cup C) \rightarrow \mathbb Z$, where $C_0$ is a line at infinity. In addition, we provide a detailed description of the collection of relations of $F$ as above in terms of the multiplicities of the roots of $\Delta_{C,\epsilon}(t)$. This generalization is made in the context of a larger class of singularities i.e. those which lead to rational orbifolds of elliptic type.
DOI：
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发表时间：
2005
期刊：
J.Math.Soc.Japan 57
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作者：
Eyral;Christophe;Oka;Mutsuo
通讯作者：
Mutsuo
DOI：
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发表时间：
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期刊：
Topology and its applications (受理)
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作者：
Eyral;C;Oka;M.
通讯作者：
M.