Dale Cutkosky has given the solution in characteristic zero of the 40-year-old Abhyankar Conjecture concerning the factorization of birational maps. Mathematicians have considered the conjecture one of the most difficult problems in math. Little progress was made toward a solution in 20 years.
Using difficult, imaginative and innovative methods, Cutkosky’s tour de force in algebraic geometry will have major consequences in the field for decades. Asked for comments on the achievement, several experts used adjectives like “spectacular, astounding, remarkable, ingenious and beautiful.”
The proof in dimension 3 is in “Local Factorization of Birational Maps,” Adv. Math 132 (1997), No. 2, 167—315, which also appeared this year as a “featured review” in Math Reviews (MR99c:14018). Reviewer Vincent Cossart says: “Theorem A is a very difficult result.... Let us say a few words about the proof, which is 146 pages long.” The proof in all dimensions is in “local monomialization and factorization of morphisms,” which will appear as a volume of Asterisque.
Shreeram S. Abhyankar is the Marshall Distinguished Professor of Mathematics at Purdue University and recipient of the Chauvenet Prize of the Mathematical Association of America. He posed the Abhyankar Conjecture in 1966 and restated it in his book Algebraic Geometry for Scientists and Engineers, Volume AMS Surveys and Monographs printed in 1990.
C.P.: Can you tell our readers, in a non-technical way, what the Abhyankar conjecture is about?
Abhyankar: The story starts with my PhD thesis, completed in 1955 under the direction of Oscar Zariski. Zariski, in the late 1940s proved that if you have two surfaces of some type and a map from one to the other, then with some conditions, such a mapping always factors through a certain “blow-up.” In my thesis and immediately following it I generalized this to dimension 2 and to more general surfaces. This led me to the general question: What happens in higher dimensions?
C.P.:What was the status of the conjecture when Dale Cutkosky solved it?
Abhyankar: I gave this problem to my various PhD students, and in 1972 one of my students named Shannon showed that exactly in the original form of Zariski, it is not true in dimension 3. I then raised the question that if one cannot do such a precise factorization, a weaker factorization ought to be true. Then a piece of this was done by another of my PhD students, Christensen, around 1979. After that, the problem remained open for many years until Dale worked on it and made this tremendous progress.