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Topology of conic bundles. (English) Zbl 0668.14013
Let \(P\to X\) be a bundle of conics on a smooth algebraic variety X which degenerates into a pair of distinct lines over a smooth irreducible divisor Y. The 2-sheeted covering of Y thus obtained defines an element \(\alpha \in H^ 1(Y,{\mathbb{Z}}/2)\). On the other hand we have a \({\mathbb{P}}^ 1\)-bundle on X-Y, and the topological obstruction to this \({\mathbb{P}}^ 1\)- bundle to be SL(2)-banal, that is, to be the projective bundle of a rank 2-topological vector bundle with trivial determinant, is an element \(\beta \in H^ 2(X-Y,{\mathbb{Z}}/2)\) (see § 1.1). Consider the Gysin map \(H^ 2(X-Y,{\mathbb{Z}}/2)\to H^ 1(Y,{\mathbb{Z}}/2)\), the composite of the coboundary map \(H^ 2(X-Y,{\mathbb{Z}}/2)\to H^ 3(X,X-Y,{\mathbb{Z}}/2)\) with the Thom isomorphism \(H^ 3(X,X-Y,{\mathbb{Z}}/2)\to H^ 1(Y,{\mathbb{Z}}/2)\), by definition.
Theorem 1. If the total space P of the conic bundle is a smooth algebraic variety then under the Gysin map, the image of the obstruction class \(\beta \in H^ 2(X-Y,{\mathbb{Z}}/2)\) is the cohomology class \(\alpha \in H^ 1(Y,{\mathbb{Z}}/2)\) defined by the 2-sheeted covering. In particular, if the 2-sheeted covering is not split, then the \({\mathbb{P}}\)-bundle on X-Y is not topologically SL(2)-banal.
Corollary 1. Under the hypothesis of theorem 1, the topological Brauer class \(\beta '\in H^ 3(X-Y,{\mathbb{Z}})\) of the \({\mathbb{P}}^ 1\)-bundle (see § 1.1) maps under the Gysin homomorphism to the Chern class \(\alpha '\in H^ 2(Y,{\mathbb{Z}})\) of the line bundle determined by the 2-sheeted cover of Y. In particular if the Chern class is not zero, then the \({\mathbb{P}}^ 1\)-bundle on X-Y is not topologically banal, that is, it is not associated to any rank-2 topological vector bundle.

14F45 Topological properties in algebraic geometry
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
57R20 Characteristic classes and numbers in differential topology
55S35 Obstruction theory in algebraic topology
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