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By Paolo (i-padv) Cupit-foutou, G. Pezzini, Math Z, Mr (j:)]. Monoids Γ, Reviewed Dmitry and A. Timashëv


Equivariant deformations of the affine multicone over a flag variety. (English summary) Adv. Math. 217 (2008), no. 6, 2800–2821. Let Γ = Z+λ1 ⊕ · · · ⊕ Z+λs be a saturated free monoid of dominant weights of a semisimple simply connected complex algebraic group G. The affine multicone Z0 over a flag variety G/P, where P = P (λ1 + · · · + λs) is the parabolic subgroup associated with λ1 + · · · + λs, is defined as the closure in V = V (λ1) ⊕ · · · ⊕ V (λs) of the G-orbit of v = vλ1 + · · · + vλs, where vλi are highest vectors in simple G-modules V (λi) of highest weights λi. The coordinate algebra of Z0 is multiplicity-free and Γ-graded: C[Z0] = ⊕ λ∈Γ V (λ)∗. The authors study equivariant deformations of Z0. It is proved that the invariant Hilbert scheme Hilb G λ, which parametrizes affine G-varieties Z with C[Z] ≃ C[Z0] as G-modules, is an affine space. The adjoint torus of G acts on Hilb G λ with the unique fixed point Z0 and the weights of TZ0HilbGλ are spherical roots of a certain wonderful G-variety X ⊆ P(V (λ1)) × · · · × P(V (λs)). The isomorphism classes of equivariant deformations Z of Z0 are parametrized by G-orbits in X so that for each x ∈ X one has Z = Gw, where w ∈ V lies over x. These results generalize those of S. Jansou [Math. Ann. 338 (2007), no. 3, 627–667; MR2317933 (2008d:14069)] for s = 1 and D. Luna [J. Algebra 313 (2007), no. 1, 292– 319; MR2326148 (2008e:14070)] for the case in which Γ is the monoid of all dominant weights. Wonderful varieties X arising in this way are characterized as those admitting a simple embeddin

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