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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 | Abstract: | We 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 | Services: | Forum | Review | PDF | Favorites |
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