Singularities of Varieties

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Deformations A deformation of a plane curve singularity F(x, y) is a germ of an analytic function G(x, y, ǫ) such that G(x, y, 0) = F(x,y). Theorem. The order of a singularity does not increase in a deformation. Problem. Which isomorphism classes arise in a deformation of a fixed singularity? Solve this problem for F(x,y) = y 2 − x n . VO AAG 18
Group Actions A Lie group is a nonsingular variety (manifold) such that the multiplication and the inverse are analytic functions. An (left) action of a Lie group G on a manifold M is an analytic function G × M → M, (g,m) ↦→ g.m, such that (gh).m = g.(h.m). The group action induces an equivalence relation on M. The equivalence classes are called orbits. VO AAG 19
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: Exercise Determine N such that x 3
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

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