E * Q = 0
back to: page on septics with many nodes
I constructed a surface of degree 7 in real 3-space admitting 99 ordinary double points (see my paper on arXiv.org: math.AG/0409348) while writing my Ph.D. thesis under the direction of Duco van Straten.
The construction is based on Rohn's idea (19th century) to globalize the local equation ((x+y)(x-y) - z2=0) of an ordinary double point, see below.
We choose the planes in a symmetric way like in Barth's construction of the 65-nodal sextic in 1996, see below.
Using the close connection to surfaces defined over finite prime fields, we were able to construct the septic with 99 nodes.
In each of the two intersection points of E:=x2-y2 = (x+y) * (x-y) with Q:=x2+y2+z2-12 the surface f:=E-Q2=0 has a singularity with local equation x2 -y2 -z2=0.
Such a singularity is called A1-singularity or node or ordinary double point. The figures below illustrate this:
W. Barth used Rohn's idea above to construct a surface of degree 6 with 65 nodes in 1996. He starts with f = E - t*Q2, where E is now the product of six planes and t a parameter. Generically, such a surface has (6*5/2) * 2 = 15 * 2 = 30 singularities at the intersection points of the 15 pairwise intersection lines of the 6 planes and the doubled quadric Q.
He chooses the six planes E to be the 6 symmetry planes of the icosahedron, which allowed him to compute a parameter t, s.t. the sextic f has 65 nodes when regarded in projective space.
Some information and images will follow, in particular on the connection to surfaces over finite prime fields and how we used them.
Until now, the only information available is my paper, sorry.
But for convenience, you can already download the Singular file that can be used to check the correctness of the result:
Oliver Labs
mail: oliver@AlgebraicSurface.net
home: www.AlgebraicSurface.net
or
Algebraic Geometry Group Mainz, Germany