Singularities of Varieties

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Weierstraß Preparation Theorem. Any plane curve singularity of order n is isomorphic to one defined by an equation F ∈ K[[x]][y] (i.e. F is polynomial in y) such that deg y (F) = n and coeff y n(F) = 1. Sketch of Proof. First, we apply an isomorphism to transform the equation F 0 of order n to another equation F 1 which is regular in y. This means that if we write F 1 as power series in y with coefficients in K[[x]], then the constant term of coeff y n(F 1 ) is not zero. This can be achieved by a linear map (x, y) ↦→ (x + αy, y) for suitable α ∈ K. Second, we write y n as F 1 Q + R with Q ∈ K[[x,y]] and R ∈ K[[x]][y], deg y (R) < n, using power series division. Then y n − R and F 1 generate the same ideal in K[[x,y]], hence y n − R is an equation as required. VO AAG 8
Exercise Prove that every double point is analytically equivalent to y 2 − x n for some n ≥ 2. VO AAG 9
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: Plane Curve Singularities A plane c
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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