2,316 research outputs found
On reductive automorphism groups of regular embeddings
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 automorphism group of X stabilizing D.
We determine a Levi subgroup A of this automorphism group, and we compute
relevant invariants of X as a spherical A-variety. As a byproduct, we obtain a
description of the open A-orbit on X and the inclusion relation between A-orbit
closures.Comment: v2: 41 pages, improved Introduction, added details in Sections 3 and
4; v3: 46 pages, minor change
Lectures on wonderful varieties
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
Automorphisms of wonderful varieties
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.Comment: 16 pages; added details and last sectio
Simple Immersions of Wonderful Varieties
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 line bundle on a
wonderful variety is very ample.Comment: LaTeX, 20 page
Rural policy lessons from OECD countries
In this paper presented at this year's rural conference, Beyond Agriculture: New Policies for Rural America, Dr. Pezzini of the OECD explored how many of the challenges facing rural America are the same challenges found in rural areas throughout the world. Although agriculture and other natural resource industries are still important economic sectors, they are creating fewer and fewer rural jobs. Rural areas suffer from the outmigration of both young and highly skilled workers, leaving an aging population and strained public services. And most rural areas have difficulty mustering the critical mass of capital and infrastructure to encourage and sustain new rural> businesses.> Pezzini noted that while countries are responding in many different ways to these challenges, successful policies appear to have three common traits. First, rural policy is shifting from a focus on individual sectors (such as farm policy) to one based on regions or territories. Second, the administration and design of such policies devolves from national governments to the "new regions," which often cut across traditional political and administrative boundaries. That is, governments are recognizing that economic regions are more meaningful than traditional policy boundaries, and attempts are being made to align the two. Third, there are new attempts to better coordinate policies affecting rural areas. At the federal level, this often involves creating new interministerial working groups.> Pezzini concluded that these policy innovations could be especially instructive to a new generation of rural policy in the United States, where farm policy has been the major focus in the past.Rural areas ; Rural development
Wonderful varieties of type D
Let G be a complex connected semisimple group, whose simple components have
type A or D. We prove that wonderful G-varieties are classified by means of
combinatorial objects called spherical systems. This is a generalization of a
known result of Luna for groups of type A; thanks to another result of Luna,
this implies also the classification of all spherical G-varieties for the
groups G we are considering. For these G we also prove the smoothness of the
embedding of Demazure.Comment: 60 pages, AMSLaTeX, 11 eps file
Orbits of strongly solvable spherical subgroups on the flag variety
Let G be a connected reductive complex algebraic group and B a Borel subgroup
of G. We consider a subgroup H of B acting with finitely many orbits on the
flag variety G/B, and we classify the H-orbits in G/B in terms of suitable
combinatorial invariants. As well, we study the Weyl group action defined by
Knop on the set of H-orbits in G/B, and we give a combinatorial model for this
action in terms of weight polytopes.Comment: v4: final version, to appear on Journal of Algebraic Combinatorics.
Apported some minor corrections to the previous versio
On the W-action on B-sheets in positive characteristic
Let G be a connected reductive group defined over an algebraically closed
ground field of characteristic p, let B be a Borel subgroup of G, and let X be
a G-variety. The first named author has shown that for p = 0 there is a natural
action of the Weyl group W on the (finite) set of closed B-invariant
subvarieties of X that are of maximal modularity, and conjectured that the same
construction yields a W-action whenever p is different from 2. In the present
paper we prove this conjecture.Comment: v2: final versio
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