Smooth Projective Symmetric Varieties With Picard Number One

Abstract

We classify the smooth projective symmetric G-varieties with Picard number one (and G semisimple). Moreover, we prove a criterion for the smoothness of the simple (normal) symmetric varieties whose closed orbit is complete. In particular we prove that, given a such variety X which is not exceptional, then X is smooth if and only if an appropriate toric variety contained in X is smooth.

Fano Symmetric Varieties with Low Rank

“…Now, we verify the smoothness of such varieties. First, we explain the conditions for a projective locally factorial symmetric variety X with rank two to be smooth (see [Ru07], Theorems 2.1 and 2.2). Let Y be a open simple Gsubvariety of X whose closed orbit is compact; then the associated colored cone (C, F ) has dimension two.…”

Effective and big divisors on a projective symmetric variety

“…In particular the standard completion of any symmetric space is Q-factorial. The conditions for the smoothness are much more complicated (see [Ru07], Theorem 2.2). Notice that the most part of this section is true for any spherical variety: in particular the descriptions of the class group and of the Picard group holds in general.…”

Normality and non-normality of group compactifications in simple projective spaces

“…To prove Theorem 3.6, we will make use of a characterization of smoothness for arbitrary group compactifications given by D. Timashev in [12]. For convenience, we will use a generalization of it which can be found in [10] in the more general context of symmetric spaces. We recall it in the case of a simple group compactification.…”

“…In [10], it is proved that when X λ is normal then Z λ also is normal. The converse of this result does not hold in general.…”