Abstract

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.