Fabrizio Catanese

Title: Nodal surfaces in 3-space and their associated binary codes

Abstract:

Nodal surfaces in 3-space are those surfaces whose singularities have nondegenerate Hessian. 
Basic numerical invariants are the degree d of such a surface Y, the number \nu of singular 
points. If you fix those integers (d,\nu) you have the so-called Severi variety F(d,\nu) of 
such nodal surfaces. The first basic question is: for which pairs is F(d,\nu) nonempty ?

The answer is known for d <= 6, in particular also the maximal number of nodes \mu (d) that a 
normal surface in 3-space of degree d can have is known only for d <= 6 (there are however 
explicit lower and upper bounds).

The maximizing nodal surfaces (those with \mu(d) nodes) are understood for d <= 5, whereas for d=6 
we know about their existence (Barth's sextic).

An invariant of normal surfaces is the fundamental group of the smooth locus, while its first homology 
is a vector space over Z/2. For maximizing nodal surfaces the former group is known for d <= 4, and 
the latter for d=5.

In general to such a nodal surface the first homology of the smooth locus determines a binary code K, 
which was used by Beauville to show that, for d=5 , \mu(d) = 31. Coding theory was crucial in order 
to prove that \mu(d) <= 65.

We shall discuss the basic notions and methods of coding theory, e.g. the McWilliams identities. 
For geometric applications, homological algebra is needed in order to determine the Hamming 
weights of vectors.

The first theorem is that for d=6 and \nu = 65 the code is uniquely determined, and can be described
explicitly via the so-called Hall graph, attached to the group \SigmaL(2, 25). The extended code K'
(considering also the class of the hyperplane section) is also uniquely described, and shows that
every 65 nodal sextic is the discriminant of the projection of a cubic hypersurface in \PP^6 with 34, 
33 or 32 nodes.

While for d <= 4 all codes associated to nodal surfaces are understood, the situation is unclear yet 
for d=5 and d=6. For d=5 geometry can exclude some codes which would be combinatorially possible.

For d=4,5,6 there is a very interesting relation with the geometry of nodal cubic hypersurfaces in 
n-space, and the linear subspaces contained in them. For instance, the maximal number of nodes 
of a cubic hypersurface is determined, and we have in dimension up to 6: 1,3,4,10,15,35. One may 
ask whether, in the case of even dimension n, the cubic hypersurface with maximal number of
singularities is projectively equivalent to the Segre cubic.

We describe the linear subspaces contained in the Segre cubic for n=6.