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On the equations for universal torsors over del Pezzo surfaces. (English) Zbl 1193.14051

The main object of the paper under review is a smooth del Pezzo surface \(X\) of degree \(d=9-r\) defined over a field \(k\) of characteristic zero. In contrast to an earlier paper of the authors [Algebra Number Theory 1, No. 4, 393–419 (2007; Zbl 1170.14026)], the surface is not necessarily split, i.e. the Galois action on the set of exceptional curves may be nontrivial. In the first part, the authors revisit the above cited paper with the aim to obtain an explicit global description of universal torsors \(\mathcal T\) over \(X\) in the split case. They start with an embedding of \(\mathcal T\) into the orbit of the highest weight vector of a fundamental representation of the simple simply connected Lie group \(G\) which has the same root system as \(X\), as in [loc. cit.]. This orbit is the punctured affine cone \(C\) over \(G/P\), where \(P\subset G\) is a maximal parabolic subgroup. The embedding is equivariant with respect to the action of the Néron–Severi torus \(T\) of \(X\), identified with a maximal torus of \(G\) extended by \(\mathbb G_{\text{m}}\). One of the main results of the paper, Theorem 2.5, provides a description of \(\mathcal T\) as the intersection of \(6-d\) dilatations of \(C\) by \(k\)-points of the maximal torus of \({\text{GL}}(V)\) which is the centralizer of \(T\) in \({\text{GL}}(V)\).
Some complementary results in the split case include a uniqueness theorem for the aforementioned embedding.
In the second part of the paper the results are extended to the nonsplit case. Assuming that \(X\) has a rational point (which is always the case for \(d=5\)), the authors construct an embedding of \(\mathcal T\) into the same homogeneous space, equivariantly with respect to the action of a possibly nonsplit torus (Theorem 4.4). This requires classification of maximal tori in quasi-split algebraic groups obtained by P. Gille [J. Ramanujan Math. Soc. 19, 213–230 (2004; Zbl 1193.20057)] and M. S. Raghunathan [J. Ramanujan Math. Soc. 19, No. 4, 281–287 (2004; Zbl 1080.20042)].

MSC:

14J26 Rational and ruled surfaces
22E46 Semisimple Lie groups and their representations
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