Constructing sextic surfaces with a given number d of nodes

Even Sets of Nodes on Sextic Surfaces

We determine the possible even sets of nodes on sextic surfaces in P, showing in particular that their cardinalities are exactly the numbers in the set {24, 32, 40, 56}. We also show that all the possible cases admit an explicit description. The methods that we use are an interplay of coding theory and projective geometry on one hand, of homological and computer algebra on the other. We give a ...

Constructing Elliptic Curves with a Given Number of Points

We describe how the theory of complex multiplication can be used to construct elliptic curves over a finite field with a given number of rational points and illustrate how this method can be applied to primality testing.

On the Number of Minimal Surfaces with a given Boundary

In [TT78], Tomi and Tromba used degree theory to solve a longstanding problem about the existence of minimal surfaces with a prescribed boundary: they proved that every smooth, embedded curve on the boundary of a convex subset of R must bound an embedded minimal disk. Indeed, they proved that a generic such curve must bound an odd number of minimal embedded disks. White [Whi89] generalized thei...

A Sextic Surface Cannot Have 66 Nodes

Sextic surfaces with ten triple points

All families of sextic surfaces with the maximal number of isolated triple points are found. Surfaces in P3(C) with isolated ordinary triple points have been studied in [EPS]. 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 [EPS] contains a complete classification. In this note I achieve the...