Singularities of Varieties

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Controlled Ideals, Singular Locus, Controlled Transform A controlled ideal is an ideal together with a positive integer. The singular locus of a controlled ideal (I,c) is the set of all points where the ideal has order at least c. If the center is contained in the singular locus, then the controlled transform is defined by dividing out the c-th power of the exceptional ideal. Resolution of controlled ideals: sequence of blowups with center in the singular locus, such that the singular locus is empty. VO AAG 42
Reduction Problems Can you reduce the resolution problem of ideals to the resolution problem of controlled ideals? Conversely? A controlled ideal is called tight iff its order is never greater then c. Can you reduce the resolution problem of controlled ideals to the resolution problem of tight controlled ideals? VO AAG 43
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 and 40: Transversality Assume H : x = 0 is
Page 41: Variants of Resolution Resolution o
Page 45 and 46: Rees Algebras Let R be the function

Singularities of Varieties

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