Steiner's conic problem

In enumerative geometry, Steiner's conic problem is the problem of finding the number (3264) of smooth conics tangent to five given conics in the complex plane in general position. It is named after Jakob Steiner who gave an incorrect solution in 1848.

Steiner (1848) claimed that the number of conics tangent to 5 given conics in general position is 7776 = 6⁵, but later realized this was wrong. The correct number 3264 was found in about 1859 by Ernest de Jonquières who did not publish because of Steiner's reputation, and by Chasles (1864) using his theory of characteristics, and by Berner in 1865. However these results, like many others in classical intersection theory, do not seem to have been given complete proofs until the work of Fulton and Macpherson in about 1978.

The space of (possibly degenerate) conics in the complex plane can be identified with the projective space P⁵. Steiner observed that the conics tangent to a given conic form a degree 6 hypersurface in P⁵. So the conics tangent to 5 given conics correspond to the intersection points of 5 degree 6 hypersurfaces, and by Bezout's theorem the number of intersection points of 5 generic degree 6 hypersurfaces is 6⁵ = 7776, which was Steiner's incorrect solution. The reason this is wrong is that the five degree 6 hypersurfaces are not in general position and have a common intersection in the Veronese surface, corresponding to the set of double lines in the plane, all of which have double intersection points with the 5 conics. In particular the intersection of these 5 hypersurfaces is not even 0-dimensional but has a 2-dimensional component. So to find the correct answer, one has to somehow eliminate the plane of spurious degenerate conics from this calculation.

One way of eliminating the degenerate conics is to blow up P⁵ along the Veronese surface. The Chow ring of the blowup is generated by H and E, where H is the total transform of a hyperplane and E is the exceptional divisor. The total transform of a degree 6 hypersurface is 6H, and Steiner calculated (6H)⁵ = 6⁵P as H⁵=P (where P is the class of a point in the Chow ring). However the number of conics is not (6H)⁵ but (6H−2E)⁵ because the strict transform of the hypersurface of conics tangent to a given conic is 6H−2E.

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