2,316 research outputs found

    On reductive automorphism groups of regular embeddings

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    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

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    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

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    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

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    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

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    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

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    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

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    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

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    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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