Singularities of Varieties

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Higher Dimensional Analogues Tougeron’s theorem is valid in any dimension. What fails to hold is finite determinacy. There exists no N such that for all F ∈ K[[x, y, z]] of order larger than N, x 2 +y 2 +F is equivalent to x 2 +y 2 . Finite determinacy does hold for isolated singularities. VO AAG 24
Singularities of Maps Let M, N be real differential manifolds (maybe with boundary). The set D(M,N) of differentiable functions admits several topologies. We can work with any topology such that the Taylor expansion at any point is continuous. Let us write D ∗ (M) for the set of all diffeomorphisms from M to itself. Then D ∗ (M) × D ∗ (N) acts on D(M, N) by (u,v).f = vfu −1 . VO AAG 25
Page 1 and 2: Singularities of Varieties Let K be
Page 3 and 4: Analytic Equivalence An analytic ma
Page 5 and 6: Smooth singularities A variety of d
Page 7 and 8: Plane Curve Singularities A plane c
Page 9 and 10: Exercise Prove that every double po
Page 11 and 12: Generalizations A hypersurface sing
Page 13 and 14: Deciding Equivalence Given two squa
Page 15 and 16: Tougerons Theorem Theorem. If G is
Page 17 and 18: Exercise Determine N such that x 3
Page 19 and 20: Group Actions A Lie group is a nons
Page 21 and 22: Application Let F(x, y) be a polyno
Page 23: Problems Which isomorphism classes
Page 27 and 28: Morse Lemma Assume N = R. Then f is
Page 29 and 30: Infinitesimal Stability Let D m,n b
Page 31 and 32: Algebraic Criterion f ∈ D m,n is
Page 33 and 34: Other Cases m = 1, n = 2: locally s
Page 35 and 36: Total transform and proper transfor
Page 37 and 38: Point Blowups in Higher Dimension T
Page 39 and 40: Transversality Assume H : x = 0 is
Page 41 and 42: Variants of Resolution Resolution o
Page 43 and 44: Reduction Problems Can you reduce t
Page 45 and 46: Rees Algebras Let R be the function

Singularities of Varieties

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