Singularities of Varieties

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Automorphisms of (K,0) A convergent power series P ∈ K[[x]], P = P 0 + P 1 x + P 2 x 2 + P 3 x 3 + . .. defines an automorphism of (K,0) iff P 0 = 0 and P 1 ≠ 0. For any power series Q ∈ K[[x]] of positive order m, Q = Q m x m + Q m+1 x m+1 +· · ·+, there is an automorphism P such that Q(P(x)) = ±x m . In case K = C, we may omit the sign. VO AAG 4
Smooth singularities A variety of dimension n is called smooth or nonsingular at some point p iff its germ at p is isomorphic to the germ of K n . The Jacobian ideal is defined as the ideal generated by all (N − n)-minors of the Jacobi matrix of a set of equations generating the vanishing ideal. (N is the number of variable, n the dimension.) Theorem. A variety is smooth at a point p iff p does not belong to the zero set of the Jacobian ideal. VO AAG 5
Page 1 and 2: Singularities of Varieties Let K be
Page 3: Analytic Equivalence An analytic ma
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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