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29 March 2024
 
  » arxiv » 1008.2018

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Mordell-Weil groups of elliptic threefolds and the Alexander module of plane curves
J.I.Cogolludo-Agustin ; A.Libgober ;
Date 11 Aug 2010
AbstractWe show that the degree of the Alexander polynomial of a plane algebraic curve with nodes and cusps as the only singularities does not exceed ${10 over 3}d-4$ where $d$ is the degree of the curve. For irreducible curves this yields the same bound of the rank of abelianization of commutator of the fundamental group of a plane curve with singularities as above. We also show that the Alexander polynomial $Delta_C(t)$ of a 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 $f^3+g^2+Fh^6$. This is obtained as a consequence of described here correspondence 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 $Delta_{C,varepsilon}(t)$ corresponding to surjections $varepsilon: pi_1(mathbb P^2- C) ightarrow mathbb Z$. 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 $Delta_{C,varepsilon}(t)$. This generalization is made in the context of a larger class of singularities i.e. those which lead to rational orbifolds of elliptic type.
Source arXiv, 1008.2018
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