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Concept# Affine variety

Summary

In algebraic geometry, an affine algebraic set is the set of the common zeros over an algebraically closed field k of some family of polynomials in the polynomial ring k[X_1, \ldots,X_n]. An affine variety or affine algebraic variety, is an affine algebraic set such that the ideal generated by the defining polynomials is prime.
Some texts call variety any algebraic set, and irreducible variety an algebraic set whose defining ideal is prime (affine variety in the above sense).
In some contexts (see, for example, Hilbert's Nullstellensatz), it is useful to distinguish the field k in which the coefficients are considered, from the algebraically closed field K (containing k) over which the common zeros are considered (that is, the points of the affine algebric set are in Kn). In this case, the variety is said defined over k, and the points of the variety that belong to kn are said k-rational or r

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We classify the spherical birational sheets in a complex simple simply-connected algebraic group. We use the classification to show that, when G is a connected reductive complex algebraic group with simply-connected derived subgroup, two conjugacy classes O-1, O-2 of G, with O-1 spherical, lie in the same birational sheet, up to a shift by a central element of G, if and only if the coordinate rings of O-1 and O-2 are isomorphic as G-modules. As a consequence, we prove a conjecture of Losev for the spherical subvariety of the Lie algebra of G. (C) 2021 Elsevier Inc. All rights reserved.

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Let G be a connected reductive algebraic group over an algebraically closed field k,gamma is an element of g( k(( epsilon ))) a semisimple regular element, we introduce a fundamental domain F gamma for the affine Springer fibers X gamma. We show that the purity conjecture of X gamma is equivalent to that of F gamma via the Arthur-Kottwitz reduction. We then concentrate on the unramified affine Springer fibers for the group GL(d). It turns out that their fundamental domains behave nicely with respect to the root valuation of gamma. We formulate a rationality conjecture about a generating series of their Poincare polynomials, and study them in detail for the group GL(3). In particular, we pave them in affine spaces and we prove the rationality conjecture.