Wonderful compactification of an arrangement of subvarieties

We construct small resolutions of the moduli space $\overline{Q}_n$ of stable scaled $n$-marked lines of Ziltener and Ma'u--Woodward and of the moduli space $\overline{P}_n$ of stable $n$-marked ${\mathbb G}_a$-rational trees introduced in earlier work. The resolution of $\overline{P}_n$ is the augmented wonderful variety corresponding to the graphic matroid of the complete graph. The resolution of $\overline{Q}_n$ is a further blowup, also a wonderful model of an arrangement...

We study the cone of effective divisors and the total coordinate ring of wonderful varieties, with applications to their automorphism group. We show that the total coordinate ring of any spherical variety is obtained from that of the associated wonderful variety by a base change of invariant subrings.

We study the family of Bethe subalgebras in the Yangian $Y(\mathfrak{g})$ parameterized by the corresponding adjoint Lie group $G$. We describe their classical limits as subalgebras in the algebra of polynomial functions on the formal Lie group $G_1[[t^{-1}]]$. In particular we show that, for regular values of the parameter, these subalgebras are free polynomial algebras with the same Poincare series as the Cartan subalgebra of the Yangian. Next, we extend the family of Bethe...

Using the wonderful compactification of a semisimple adjoint affine algebraic group G defined over an algebraically closed field k of arbitrary characteristic, we construct a natural compactification Y of the G-character variety of any finitely generated group F. When F is a free group, we show that this compactification is always simply connected with respect to the \'etale fundamental group, and when k=C it is also topologically simply connected. For other groups F, we desc...

Some projective wonderful models for the complement of a toric arrangement in a n-dimensional algebraic torus T were constructed in [3]. In this paper we describe their integer cohomology rings by generators and relations.

De Concini-Procesi introduced varieties known as wonderful compactifications, which are smooth projective compactifications of semisimple adjoint groups $G$. We study the Frobenius pushforwards of invertible sheaves on the wonderful compactifications, and in particular its decomposition into locally free subsheaves. We give necessary and sufficient conditions for a specific line bundle to be a direct summand of the Frobenius pushforward of another line bundle, formulated in t...

For any triple given by a positive integer n, a finite group G, and a faithful representation V of G, one can describe a subspace arrangement whose intersection lattice is a generalized Dowling lattice in the sense of Hanlon. In this paper we construct the minimal De Concini-Procesi wonderful model associated to this subspace arrangement and give a description of its boundary. Our aim is to point out the nice poset provided by the intersections of the irreducible components i...

Given a subspace arrangement, there are several De Concini-Procesi models associated to it, depending on distinct sets of initial combinatorial data (building sets). The first goal of this paper is to describe, for the root arrangements of types A_n, B_n (=C_n), D_n, the poset of all the building sets which are invariant with respect to the Weyl group action, and therefore to classify all the wonderful models which are obtained by adding to the complement of the arrangement a...

Wonderful compactifications of adjoint reductive groups over an algebraically closed field play an important role in algebraic geometry and representation theory. In this paper, we construct an equivariant compactification for adjoint reductive groups over arbitrary base schemes. Our compactifications parameterize classical wonderful compactifications of De Concini and Procesi as geometric fibers. Our construction is based on a variant of the Artin-Weil method of birational g...

A geometrical realization of wonderful varieties by means of a suitable class of invariant Hilbert schemes is given. As a consequence, Luna's conjecture asserting that wonderful varieties are classified by spherical systems, triples of combinatorial invariants, is proved.