Singularities of Varieties

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Example m = n = 1. We may assume that f(x) = x d for some d > 0. Then the image of the differential of right equivalence together with translation is the ideal generated by f ′ (x), which is the vectorspace generated by x a , a ≥ d − 1. The ideal generated by left equivalence is the vectorspace generated by x b , d | b. It follows that x d is stable iff d = 1 or d = 2. VO AAG 30
Algebraic Criterion f ∈ D m,n is infinitesimally stable iff the the quotient of R[[x 1 ,...,x m ]] n by the module generated by the partial derivatives and by f i e j , where f i is the i-th coordinate of f and e j is the j-th unit vector, is generated by the unit vectors over R. Example (x, y) ↦→ (x 3 + xy, y). VO AAG 31
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 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: Infinitesimal Stability Let D m,n b
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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