By [various contributors], Selman Akbulut (Michigan State University), Turgut Onder (Middle East Technical University), Ronald J. Stern (University of California at Irvine)
Devoted to the reminiscence of Raoul Bott, a superb mathematician of the 20 th Century, this quantity includes articles from either 11th and 12th Gokova meetings, held in Gokova, Turkey.
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Now if Z ⊂ R4 were an algebraic hyper-surface, we can choose a square free polynomial equation f (x) = 0 of Z. Say, f takes signs ± on the inside 54 Real algebraic structures and outside regions B± ⊂ R4 separated by S 3 ⊂ Z. Define R4 := {(x, t) ∈ R4 × R | t2 = f (x)} ⊃ Z ⊃ V Then χ(lkx (V, R4 )) − χ(lkx (V, R4 )) = χ(lkx (V, B+ )) − χ(lkx ((V, B− )) mod 4 is not generically constant (it is 2 or 0 when x is in the interior of I or I ′ ), this violates (*). 4. Transcendental manifolds If we ask whether a smooth submanifold M ⊂ Rn is isotopic to a nonsingular real algebraic set V in a strong sense, then we can find genuine obstructions to doing this, even when M is already nonsingular algebraic set in Rn .
Conclusion: In ambient dimension 1, there arise no equiconstant points. Example 3: What about dimension 2, say plane curves? We take again xm − y k = 0 with k ≥ m, of order m at 0. We blow up the origin. If k < 2m, the order drops at all points of E ∼ = P1 . If k ≥ 2m there is precisely one point in E where the order remains constant, namely the origin of the y-chart. There, the equation of the weak transform is xm − y k−m = 0. The exponent of the second monomial has dropped. How to profit of this drop?
Encouraged by the perspective of nice commutative diagrams we apply induction on the dimension to show that we can resolve I− inside H by a sequence of blowups with centers in H. If we are lucky, the centers are also contained in top(I) so as not to increase the order of I. Stop! The ideal I− does not pass in this process to its strict or weak transform, but to its controlled transform, and, as we have seen many times, its order may increase. 5 introductory examples of this chapter. There, the ideal I− corresponded to the monomial y α , and we saw that once I− is a monomial (if defined correctly), the drop of the order of I can be forced by choosing centers of maximal dimension.