Wonderful compactification

In algebraic group theory, a wonderful compactification of a variety acted on by an algebraic group ${\displaystyle G}$ is a ${\displaystyle G}$-equivariant compactification such that the closure of each orbit is smooth. Corrado de Concini and Claudio Procesi (1983) constructed a wonderful compactification of any symmetric variety given by a quotient ${\displaystyle G/G^{\sigma }}$ of an algebraic group ${\displaystyle G}$ by the subgroup ${\displaystyle G^{\sigma }}$ fixed by some involution ${\displaystyle \sigma }$ of ${\displaystyle G}$ over the complex numbers, sometimes called the De Concini–Procesi compactification. Elisabetta Strickland (1987) generalized this construction to arbitrary characteristic. In particular, by writing a group ${\displaystyle G}$ itself as a symmetric homogeneous space, ${\displaystyle G=(G\times G)/G}$ (modulo the diagonal subgroup), this gives a wonderful compactification of the group ${\displaystyle G}$ itself.