Singularities of Varieties

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The Vanishing Ideal Let R := K[[x 1 , ...,x n ]] be the ring of convergent power series. Let V ⊆ U ⊆ K n be a singularity. Then Ideal(V ) is defined as the ideal of all f ∈ R that vanish at all points in V . It does not depend on the choice of the representative. Any ideal of R is finitely generated. Vanishing ideals are radical. The converse holds for the case K = C, but not for the case K = R. VO AAG 2
Analytic Equivalence An analytic map from a variety X to a variety Y is a function f : X → Y whose coordinates are analytic functions. Two varieties are analytically equivalent iff there are analytic maps f : X → Y and g : Y → X such that fg = id Y and gf = id X . Two singularities are analytically equivalent iff there exists an analytic isomorphism between representatives taking o to o. Two singularities V 1 and V 2 are analytically equivalent iff their quotient algebras R 1 /Ideal(V 1 ) and R 2 /Ideal(V 2 ) are isomorphic as K-algebras. VO AAG 3
Page 1: Singularities of Varieties Let K be
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 and 24: Problems Which isomorphism classes
Page 25 and 26: Singularities of Maps Let M, N be r
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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