Singularities of Varieties

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Group Actions Theorem. Let N ⊆ N be a submanifold. Let p ∈ N. If the tangents space of N and the image of the tangent space of G under the differential of the map G → M, g ↦→ g.p, generate the tangent space of M, then N intersects every orbit in a neighborhood of p. VO AAG 20
Application Let F(x, y) be a polynomial, and let n be an integer such that F(x,y) + G(x, y) ≡ F(x, y) for any G or order greater than n. We consider the group G 1 × G 2 , where G 1 is the group of all analytic isomorphisms truncated at order n, and G 2 is the group of all units in the power series ring, truncated at order n. The set M is the set of all power series truncated at order n, G 1 acts by substitution and G 2 by multiplication. VO AAG 21
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: Group Actions A Lie group is a nons
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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