Singularities of Varieties

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Blowup of Submanifolds Let o ∈ C n . Then the blowup the submanifold L = {o} × C m ⊂ C n+m is just the direct product of the blowup of C n at o with C m . It can be covered by n open subsets isomorphic to C n × C m . The exceptional hypersurface is p −1 (L). Example: blowup x = y = 0 in C 3 . VO AAG 38
Transversality Assume H : x = 0 is an exceptional divisor in C 3 . We blow up the curve C : y = x + z 2 = 0. Step 1: Coordinate transformation making the curve into a line (x is replaced by x − z 2 ): H : x − z 2 = 0, C : y = x = 0. Step 2: First chart of blowup: x is replaced by xy. exceptional divisor becomes singular. Then the “old” VO AAG 39
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: Point Blowups in Higher Dimension T
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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