Fano symmetric varieties with low rank
A complex Gorenstein normal projective algebraic variety is called Fano if the anticanonical divisor is ample, it is called quasi-Fano if the anticanonical divisor is big and nef. The paper under review contains several results characterising the properties of being Fano and quasi-Fano for some classes of projective normal symmetric varieties.NEWLINENEWLINELet \(G\) be a reductive group and \(B\) a Borel subgroup of \(G\). A symmetric variety is a normal \(G\)-variety \(X\) possessing an open orbit \(G/H\) such that \(G^\theta \subset H \subset N(G^\theta)\), where \(\theta : G \to G\) is an involution, \(G^\theta\) is the set of fixed points of \(\theta\) and \(N(G^\theta)\) is the normaliser of \(G^\theta\) in \(G\). The rank of \(X\) is the rank of the lattice of \(B\)-weights in the field of rational functions of \(X\). By the work of \textit{D. N. Akhiezer} [Ann. Global Anal. Geom. 1, No. 1, 49--78 (1983; Zbl 0537.14033)], it follows that every normal projective symmetric variety of rank one is smooth and Fano. In the present paper it is given a classification of the smooth (resp. locally factorial) symmetric varieties of rank two which are Fano and, in case \(G\) is semisimple, also of those which are only quasi-Fano.NEWLINENEWLINEIn case \(H = N(G^\theta)\), a wonderful compactification of \(G/H\) was introduced by \textit{C. De Concini} and \textit{C. Procesi} [``Complete symmetric varieties'', Lect. Notes Math. 996, 1--44 (1983; Zbl 0581.14041)]. The author shows that such wonderful compactifications are always quasi-Fano, and those which are Fano are classified. Then they are classified the Fano symmetric varieties which are obtained from a wonderful compactification by a sequence of blow-ups along closed orbits. In the rank three case, it is also given a classification of the Fano symmetric varieties which are obtained from a wonderful compactification by a sequence of blow-ups along \(G\)-stable subvarieties.
scientific article; zbMATH DE number 6054391