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Effective divisor classes of a projective plane bundle over an elliptic curve. (English) Zbl 1428.14059

Let \(B\) be an elliptic curve. The monoid \(\mathcal{M}\) of effective divisor classes in the Neron-Severi group of \(\mathbb{P}^2\)-bundles \(W=\mathbb{P}_B(E)\) over \(B\) is investigated. Since the isomorphism class of \(W\) is determined by the vector bundle \(E\) up to the twist by a line bundle, one can assume that \(E\) is normalized, in the sense that \(h^0(E)\not=0\) but \(h^0(E \otimes L)=0\) for any line bundle \(L\) on \(B\) of negative degree. Denote by \(\pi:W \to B\) the projection, by \(T\) and \(F\) the tautological divisor and a fiber of \(\pi\) respectively, and let \(\mathcal{E}_B(r,d)\) be the set of isomorphism classes of indecomposable vector bundles of rank \(r\) and degree \(d\) on \(B\). According to a result of J. Rosoff, who investigated \(\mathcal{M}\) for \(\mathbb{P}^1\)-bundles over a curve of any genus [Pac. J. Math. 202, No. 1, 119–124 (2002; Zbl 1054.14050)], when \(E\) has rank \(2\) the classes of \(T\) and \(F\) generate \(\mathcal{M}\) except when \(E\in\mathcal{E}_B(2,1)\), in which case there is one more generator, algebraically equivalent to \(2T-F\). When \(E\) has rank 3, extending Rosoff argument, the author determines the generators of \(\mathcal{M}\) for each of the isomorphism classes of \(W\). In several cases the classes of \(T\) and \(F\) are enough to generate \(\mathcal{M}\); in some other cases one more generator is needed, which is algebraically equivalent to either \(2T-F\) or \(3T-F\), while in the remaining case, which corresponds to \(E \in \mathcal{E}(3,1)\), \(\mathcal{M}\) is generated by the classes of \(T\), \(F\), and of two further effective divisors which are algebraically equivalent to \(2T-F\) and \(3T-F\) respectively. Next the author studies the structure and the numerical characters of smooth surfaces algebraically equivalent to multiples of a generator of \(\mathcal{M}\), different from \(T\) and \(F\), when \(E \in\mathcal{E}_B(3,d)\), with \(d=1\) or \(2\). Furthermore, relying on this study, he classifies the \(\mathbb{P}^2\)-bundles \(W\) as above containing a smooth properly elliptic surface \(S\) with \(\chi(\mathcal{O}_S)=0\).

MSC:

14H60 Vector bundles on curves and their moduli
14C20 Divisors, linear systems, invertible sheaves
14J27 Elliptic surfaces, elliptic or Calabi-Yau fibrations
14J30 \(3\)-folds

Citations:

Zbl 1054.14050
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References:

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