Singularities of Varieties

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Stable Maps A map f : M → N is stable iff its orbit under the action above is open in D(M,N). This means that any sufficiently close map is equivalent to it. Example. M = N = [−1,1]. Characterize all stable maps. In this example, the stable maps are dense, i.e. any map can be deformed to a stable map. VO AAG 26
Morse Lemma Assume N = R. Then f is stable iff the Hessian is nondegenerate at all critical points, and the function values <strong>of</strong> the critical points ar all different. Moreover, the set <strong>of</strong> stable functions is dense. VO AAG 27
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: Singularities of Maps Let M, N be r
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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