Singularities of Varieties

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Transversality Better: only blowup centers which intersect all old exceptional hypersurfaces transversally. This means: for any point, there is an analytic coordinate system such that the center is defined by coordinates AND all exceptional hypersurfaces are defined by coordinates. VO AAG 40
Variants of Resolution Resolution of varieties: sequence of blowups such that the proper transform is nonsingular. Resolution of functions: sequence of blowups such that the pullback can locally be defined by products of coordinates (monomials). Resolution of ideals: sequence of blowups such that the pullback is locally generated by such a monomial. Resolution of hypersurfaces can be reduced to resolution of functions. Resolution of varieties and resolution of functions can be reduced to resolution of ideals. VO AAG 41
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 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: Transversality Assume H : x = 0 is
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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