Calculating the Mordell-Weil rank of elliptic threefolds and the cohomology of singular hypersurfaces
There is an arithmetic part to this paper, and a more purely geometric part. The latter is motivated by the former but is of independent interest. The arithmetic question is to compute the rank of the Mordell-Weil group \(\text{MW}(\pi)\) of an elliptic 3-fold \(\pi: X\to S\) where the base \(S\) is a smooth rational surface, i.e.\ of an elliptic curve over the function field \({\mathbb C}(S)={\mathbb C}(s_1,s_2)\). The method uses the fact that \(X\) has a model \(Y\), given by a minimal Weierstrass equation as a singular hypersurface in a weighted projective space. As long as \(H^4(Y)\) has a pure weight~\(4\) Hodge structure, which is only a very mild condition, the authors prove that \[ \text{rank}\text{MW}(\pi)=\text{rank}\big(H^{2,2}(H^4(Y{\mathbb C}))\cap H^4(Y,{\mathbb Z})\big)-1. \] If \(X\) is general then \(Y\) is quasismooth and in that case \(\text{MW}(\pi)=0\). The authors give explicit conditions on the discriminant under which this happens, and explain why the problem is harder for elliptic surfaces The second part of the paper aims to compute the cohomology of a hypersurface in weighted projective space, under conditions on the singularities that \(Y\) might often satisfy. The authors extend the methods used by Dimca for hypersurfaces with isolated semi-weighted homogeneous singularities, so as to compute \(H^4(Y)\) provided that the codimension~\(2\) singularities are of transverse Du Val type and the ccodimension \(3\) singularities are contact equivalent to weighted homogeneous singularities: this is quite hard. They are able to verify that these conditions are satisfied by the models of many elliptic threefolds, and thus to compute their Mordell-Weil ranks.
Mordell-Weil group of elliptic threefolds
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