Singularities of Varieties

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Examples F(x, y) = x 3 + y 3 , M = {x 3 + y 3 + a ij x i y j | 1 ≤ i + j ≤ 5,a ij ∈ K}, TM = {a ij x i y j | 1 ≤ i + j ≤ 5}, G 1 = {(u ij x i y j ,v ij x i y j ) | 1 ≤ i + j ≤ 5,u 10 v 01 − u 01 v 10 ≠ 0} , TG 1 = {(u ij x i y j , v ij x i y j ) | 1 ≤ i + j ≤ 5} , G 2 = {w ij x i y j | 0 ≤ i+j ≤ 5, w 00 ≠ 0}, TG 2 = {w ij x i y j | 0 ≤ i+j ≤ 5}, (TG 1 × TG 2 ) → TM, (U,V,W) ↦→ 3x 2 U + 3y 2 V + (x 3 + y 3 )W The image of the map is the ideal generated by δ x F,δ y F,F. Any vectorspace in TM that contains representatives for all residue classes satisfies the assumption of the theorem. VO AAG 22
Problems Which isomorphism classes arise in deformations of x 3 + y 3 ? Same question for x 2 y + y 4 , x 3 + y 4 , x 3 + y 5 , x 3 + y 6 . VO AAG 23
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: Application Let F(x, y) be a polyno
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 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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