# Guido PezziniSapienza University of Rome | la sapienza · Department of Mathematics "Guido Catelnuovo"

Guido Pezzini

Ph.D. Mathematics

## About

18

Publications

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174

Citations

## Publications

Publications (18)

Given a connected simply connected semisimple group G and a connected spherical subgroup K we determine the generators of the extended weight monoid of G/K, based on the homogeneous spherical datum of G/K. Let H be a reductive subgroup of G and let P be a parabolic subgroup of H for which G/P is spherical. A triple (G,H,P) with this property is cal...

We apply the combinatorial theory of spherical varieties to characterize the momentum polytopes of polarized projective spherical varieties. This enables us to derive a classification of these varieties, without specifying the open orbit, as well as a classification of all Fano spherical varieties. In the setting of multiplicity free compact and co...

We apply the combinatorial theory of spherical varieties to characterize the momentum polytopes of polarized projective spherical varieties. This enables us to derive a classification of these varieties, as well as a classification of all Fano spherical varieties. While considering the more general setting of multiplicity free and compact Hamiltoni...

These notes are an introduction to wonderful varieties. We discuss some general results on their geometry, their role in the theory of spherical varieties, several aspects of the combinatorics arising from these varieties, and some examples.

We define and study the global Okounkov moment cone of a projective spherical variety X, generalizing both the global Okounkov body and the moment body of X defined by Kaveh and Khovanskii. Under mild assumptions on X we show that the global Okounkov moment cone of X is rational polyhedral. As a consequence, also the global Okounkov body of X, with...

Let G be a complex connected reductive group. I. Losev has shown that a smooth affine spherical G-variety X is uniquely determined by its weight monoid, which is the set of irreducible representations of G that occur in the coordinate ring of X. In this paper we use a combinatorial characterization of the weight monoids of smooth affine spherical v...

Let G G be a connected reductive group, and let X X be a smooth affine spherical G G -variety, both defined over the complex numbers. A well-known theorem of I. Losev’s says that X X is uniquely determined by its weight monoid, which is the set of irreducible representations of G G that occur in the coordinate ring of X X . In this paper, we use th...

Let G G be a connected reductive group defined over an algebraically closed base field of characteristic p ≥ 0 p\ge 0 , let B ⊆ G B\subseteq G be a Borel subgroup, and let X X be a G G -variety. We denote the (finite) set of closed B B -invariant irreducible subvarieties of X X that are of maximal complexity by B 0 ( X ) \mathfrak {B}_{0}(X) . The...

Let G be a connected reductive complex algebraic group and B a Borel subgroup of G. We consider a subgroup H⊂B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H \subset...

We define and study a class of spherical subgroups of a Kac-Moody group. In
analogy with the standard theory of spherical varieties, we introduce a
combinatorial object associated with such a subgroup, its homogeneous spherical
datum, and we prove that it satisfies the same axioms as in the
finite-dimensional case.
Our main tool is a study of varie...

Let G be a semisimple complex algebraic group, and H ⊆ G a wonderful subgroup. We prove several results relating the subgroup H to the properties of a combinatorial invariant S of G/H, called its spherical system. It is also possible to consider a spherical system S as a datum defined by purely combinatorial axioms, and under certain circumstances...

Let G be a connected reductive complex algebraic group acting on a smooth
complete complex algebraic variety X. We assume that X under the action of G is
a regular embedding, a condition satisfied in particular by smooth toric
varieties and flag varieties. For any set D of G-stable prime divisors, we
study the action on X of the connected automorph...

We provide the spherical systems of the wonderful reductive subgroups of any
reductive group.

We complete the classification of wonderful varieties initiated by D. Luna.
We review the results that reduce the problem to the family of primitive
varieties, and report the references where some of them have already been
studied. Finally, we analyze the rest case-by-case.

Let G be an adjoint semisimple complex linear group. We analyze the
relationship between the algebraic and the combinatorial point of view on
Luna's theory of wonderful G-varieties. Some techniques relating a spherical
system to its associated wonderful subgroup are discussed; as a byproduct, we
reduce for any group G the proof of the classificatio...

Let G be a complex semisimple linear algebraic group, and X a wonderful G-variety. We determine the connected automorphism group of X and we calculate Luna's invariants of X under its action.

Let G be a semisimple connected linear algebraic group over C, and X a wonderful G-variety. We study the possibility of realizing X as a closed subvariety of the projective space of a simple G-module. We describe the wonderful varieties having this property as well as the linear systems giving rise to such immersions. We also prove that any ample l...

We show that the proof of Luna’s conjecture about the classification of general wonderful G-varieties can be reduced to the analysis of finitely many families of primitive cases. We work out all primitive cases arising with any classical group G. Luna’s conjecture states that, for any semisimple complex algebraic group G, wonderful G-varieties are...

## Projects

Project (1)

The goal of the project is to study various aspects of strongly solvable spherical subgroups in reductive algebraic groups using their structure theory that admits a combinatorial description in terms of root systems. Applications concern studying diverse geometric properties of the corresponding spherical homogeneous spaces in terms of this combinatorial description.