Effective and big divisors on a projective symmetric variety
Let \(G\) be a semisimple connected algebraic group over an algebraically closed field of characteristic zero and let \(\theta\) be an involution of \(G\) with set of fixed points \(G^\theta\). Let \(H\) be a subgroup of the normalizer of \(G^\theta\) which contains \(G^\theta\) and let \(X\) be a normal projective embedding of the symmetric variety \(G/H\). In the present paper, the author studies the effective cone and the big cone of \(X\), and a combinatorial criterion for the bigness of a nef divisor on \(X\) is given as well.NEWLINENEWLINEWhen \(X\) is \(\mathbb{Q}\)-factorial (i.e. when the rational Picard group equals the rational Weil divisor class group), an explicit description of the extremal rays of the effective cone is given, and an explicit description of the big cone is also deduced. The effective cone is described then in the non \(\mathbb{Q}\)-factorial case as the intersection of the real Picard group with an explicitly described polyhedral cone in the real Weil divisor class group.NEWLINENEWLINEFix a maximal torus \(T\) of \(G\) and a Borel subgroup \(B\) containing \(T\). The last part of the paper is devoted to characterize combinatorially the bigness of a nef Cartier divisor \(D\) on \(X\), which up to linear equivalence may always be assumed \(B\)-stable: the criterion expresses the bigness of \(D\) in terms of the \(T\)-weights of the fibers of the associated line bundle \(\mathcal O(D)\) over the \(B\)-fixed points in \(X\). This is essentially done by reducing to the case where \(X\) is toroidal, i.e. when every \(B\)-stable prime divisor of \(X\) which contains a \(G\)-orbit is \(G\)-stable. Under this assumption, there exists a projective variety \(Z \subset X\) which is toric under the action of \(T\) and which uniquely determines \(X\) as a compactification of \(G/H\). A main step in the proof of the criterion in the toroidal case is the fact that the restriction to \(Z\) of a big divisor on \(X\) is still big.
scientific article; zbMATH DE number 6078359
scientific article