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Schubert varieties and representations of Dynkin quivers

Grzegorz Bobiński, Grzegorz Zwara (2002)

Colloquium Mathematicae

We show that the types of singularities of Schubert varieties in the flag varieties Flagₙ, n ∈ ℕ, are equivalent to the types of singularities of orbit closures for the representations of Dynkin quivers of type 𝔸. Similarly, we prove that the types of singularities of Schubert varieties in products of Grassmannians Grass(n,a) × Grass(n,b), a, b, n ∈ ℕ, a, b ≤ n, are equivalent to the types of singularities of orbit closures for the representations of Dynkin quivers of type 𝔻. We also show that...

SL 2 -equivariant polynomial automorphisms of the binary forms

Alexandre Kurth (1997)

Annales de l'institut Fourier

We consider the space of binary forms of degree n ≥ 1 denoted by R n : = ℂ [ x , y ] n . We will show that every polynomial automorphism of R n which commutes with the linear SL 2 ( ℂ ) -action and which maps the variety of forms with pairwise distinct zeroes into itself, is a multiple of the identity on R n .

Smallness problem for quantum affine algebras and quiver varieties

David Hernandez (2008)

Annales scientifiques de l'École Normale Supérieure

The geometric small property (Borho-MacPherson [2]) of projective morphisms implies a description of their singularities in terms of intersection homology. In this paper we solve the smallness problem raised by Nakajima [37, 35] for certain resolutions of quiver varieties [37] (analogs of the Springer resolution): for Kirillov-Reshetikhin modules of simply-laced quantum affine algebras, we characterize explicitly the Drinfeld polynomials corresponding to the small resolutions. We use an elimination...

Spherical roots of spherical varieties

Friedrich Knop (2014)

Annales de l’institut Fourier

Brion proved that the valuation cone of a complex spherical variety is a fundamental domain for a finite reflection group, called the little Weyl group. The principal goal of this paper is to generalize this theorem to fields of characteristic unequal to 2. We also prove a weaker version which holds in characteristic 2, as well. Our main tool is a generalization of Akhiezer’s classification of spherical varieties of rank 1.

Stability of pencils of plane quartic curves

Edoardo Ballico, Paolo Oliverio (1983)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

In questa nota si danno dei criteri per la stabilità di fasci di quartiche piane.

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