Complete symmetric varieties.

*(English)*Zbl 0581.14041
Invariant theory, Proc. 1st 1982 Sess. C.I.M.E., Montecatini/Italy, Lect. Notes Math. 996, 1-44 (1983).

[For the entire collection see Zbl 0511.00010.]

Let \(G\) be a semisimple adjoint group, \(\sigma\) an automorphism of order \(2\) of \(G\), and \(H=G^{\sigma}\). The authors construct (in a canonical way) an algebraic variety \(X\) together with an action of \(G\) on \(X\) such that: (1) \(X\) has an open orbit isomorphic to \(G/H\); (2) \(X\) is smooth with finitely many orbits; (3) the orbit closures are all smooth, and also satisfying some further good properties.

This construction should be regarded as a generalization of the variety \(X\), which is the closure in \(P^ 2\times \check P^ 2\) of the set of all pairs \((C,C')\) consisting of a plane conic \(C\) and its dual \(C'\), as well as of the variety arising from the corresponding generalization to higher-dimensional quadrics. Such kind of varieties have important significance in problems of enumerative geometry. Then the authors study thoroughly these varieties by computing their Picard groups, by determining their positive line bundles and computing their cohomology, or by giving precise algorithms in order to compute the so-called characteristic numbers.

As an illustration of the general results they obtain, they reconsider the classical example due to H. Schubert for the space quadrics and compute the number of quadrics which are tangent to nine quadrics in general position.

Let \(G\) be a semisimple adjoint group, \(\sigma\) an automorphism of order \(2\) of \(G\), and \(H=G^{\sigma}\). The authors construct (in a canonical way) an algebraic variety \(X\) together with an action of \(G\) on \(X\) such that: (1) \(X\) has an open orbit isomorphic to \(G/H\); (2) \(X\) is smooth with finitely many orbits; (3) the orbit closures are all smooth, and also satisfying some further good properties.

This construction should be regarded as a generalization of the variety \(X\), which is the closure in \(P^ 2\times \check P^ 2\) of the set of all pairs \((C,C')\) consisting of a plane conic \(C\) and its dual \(C'\), as well as of the variety arising from the corresponding generalization to higher-dimensional quadrics. Such kind of varieties have important significance in problems of enumerative geometry. Then the authors study thoroughly these varieties by computing their Picard groups, by determining their positive line bundles and computing their cohomology, or by giving precise algorithms in order to compute the so-called characteristic numbers.

As an illustration of the general results they obtain, they reconsider the classical example due to H. Schubert for the space quadrics and compute the number of quadrics which are tangent to nine quadrics in general position.

Reviewer: L.Bădescu

##### MSC:

14N10 | Enumerative problems (combinatorial problems) in algebraic geometry |

14L30 | Group actions on varieties or schemes (quotients) |

14L24 | Geometric invariant theory |