Published online by Cambridge University Press: 11 November 2009
Abstract
All families of sextic surfaces with the maximal number of isolated triple points are found.
Evaluation of the conditions imposed by ten triple points requires the solution of complicated systems of equations. Thanks to Gert-Martin's efforts the computer algebra system Singular [3] is around, making such computations possible.
Surfaces in ℙ3(ℂ) with isolated ordinary triple points have been studied in [2]. The results are most complete for degree six. A sextic surface can have at most ten triple points, and such surfaces exist. For up to nine triple points [2] contains a complete classification. In this note I achieve the same for ten triple points.
The study of sextics with nine triple points is easier, because they do lie on a quadric Q. Given such a sextic with equation F the general element of the pencil αF + βQ3 is again a sextic with nine isolated triple points. It turns out that such a pencil also contains reducible surfaces, which are much easier to construct. The same argument shows that a sextic with ten triple points is a degeneration of one with nine (simply choose a quadric through nine of the ten points).
Therefore one can look for sextics with ten triple points in each of the five families given in [2]. In fact it surfaces to consider only those two, which have a rather nice description. The one-parameter family of examples [2] was found in the first family by imposing extra symmetry.
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