Singularities of Varieties

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Reducible Singularities An irreducible singularity is one which is not a union of two strictly smaller singularities. Example: The zero set of y 2 − x 2 − x 3 at the origin is the union of two smooth analytic curve germs, namely the zero set of y + x + 1 2 x2 − 1 8 x3 + 1 16 x4 − 5 128 x5 . .. and the zero set of y − x − 1 2 x2 + 1 8 x3 − 1 16 x4 + 5 128 x5 ... . VO AAG 6
Plane Curve Singularities A plane curve singularity is given by a squarefree power series F(x,y) vanishing at o. The order of F is invariant under analytic isomorphisms. The singularity is smooth iff ord(F) = 1. If a plane curve singularity is reducible, then its order is equal to the sum of orders of the components. VO AAG 7
Page 1 and 2: Singularities of Varieties Let K be
Page 3 and 4: Analytic Equivalence An analytic ma
Page 5: Smooth singularities A variety of d
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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