Singularities of Varieties

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Blowing Up The blowup of the plane C 2 is an algebraic map p : Y → C 2 , where Y is a manifold that can be covered by two open subsets isomorphic to C 2 . It has the following properties: p −1 (o) is a curve E ⊂ Y. Apart from that, p restricts to an isomorphism Y − E → C 2 − {o}. The points of E correspond to tangent directions at o. Any automorphism of C 2 that fixes o lifts to an automorphism of Y preserving E. Even more, any analytic automorphism of the germ of C 2 at o lifts to a local automorphism of a neighborhood of E in Y fixing E. VO AAG 34
Total transform and proper transform If C ⊂ C 2 is a curve containing o, then p −1 (C) has two components: the proper transform ˜C and the exceptional curve E. The pullback of the equation F of C factors into x m ˜f(x, ỹ), where ˜f is the local equation of ˜C, x is the equation of the exceptional curve, and m is the order of C at o. The curve ˜C may have “new” singular points in E. The sum of all orders at these new points is less than or equal to m. Theorem. The procedure of blowing up singular points is finite. VO AAG 35
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: Other Cases m = 1, n = 2: locally s
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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