Varieties of nodal surfaces, Coding theory, and cubic discriminants

Abstract: Nodal Hypersurfaces Y in projective space are those whose singularities have nondegenerate Hessian. Basic numerical invariants are the dimension n and the degree d of the hypersurface Y, and the number \nu of singular points. If you fix those integers (n, d,\nu) these hypersurfaces are parametrized by the so-called Nodal Severi varieties F(n, d, \nu). The first basic questions concerning them are: 1) for which triples is F(n, d, \nu) nonempty ? 2) When is it irreducible ?

Already intriguing is the situation for surfaces, indeed for n=2 the answer to 1) is known for d <= 6, also the maximal number of nodes \mu (d) that a nodal surface in 3-space of degree d can have is known only for d <= 6.

The known maximizing nodal surfaces (those with \mu(d) nodes) are: the Cayley cubic, the Kummer quartic surfaces, the Togliatti quintics, the Barth sextic.

An important chapter in Coding theory is the theory of binary linear codes, vector subspaces of a vector space (Z/2)^n.

I will recall basic notions and methods of coding theory (e.g. the McWilliams identities) and describe some codes related to quadratic forms.

Nodal surfaces are related to coding theory via the first homology of their smooth part: it is a binary code K, which was used by Beauville to show that, for d=5 , \mu(d) = 31. Coding theory was also crucial in order to prove that \mu(6) < = 65.

Our main results concern the cases d = 4,5,6 (d=2,3 being elementary).

THM 1. For d=4 the components of F(4, \nu) and their incidence correspondence are determined by their extended codes K’, which are all the shortenings of the first Reed Muller code.

We extend this result to nodal K3 surfaces of all degrees, this sheds light on the case d=5.

THM 2. For d=5 the codes K occurring are classified, up to a possible exception. F(5, \nu) is irreducible for \nu = 31, and for \nu = 29,30,31 these surfaces are discriminants of the projection of a cubic hypersurface in 5-space.

THM 3. For d=6 and \nu = 65 the codes K, K’ are uniquely determined, and can be described explicitly via the Doro-Hall graph, attached to the group \SigmaL(2, 25), and the geometry of the Barth sextic. Every 65 nodal sextic occurs as discriminant of the projection of a cubic hypersurface in 6-space with < = 33 nodes.

Irreducibility for d=6, and 65 nodes, is related to the geometry of nodal cubic hypersurfaces in n-space, and of the linear subspaces contained in them.

We pose the question whether, in the case of even dimension n, the cubic hypersurface with maximal number of singularities is projectively equivalent to the Segre cubic s_1=s_3=0 (which is locally rigid).

For theorem 2 I benefited of the cooperation of Sandro Verra, for theorem 3 of Yonghwa Cho, Michael Kiermaier, Sascha Kurz and the Linux Cluster of the Universitaet Bayreuth, while Davide Frapporti and Stephen Coughlan cooperated for the geometry of nodal cubic hypersurfaces.

algebraic geometry

Audience: researchers in the topic

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ZAG (Zoom Algebraic Geometry) seminar

Series comments: Description: ZAG seminar

The seminar takes place on Tuesdays and Thursdays via Zoom. Zoom passwords are given via mailing list on Fridays. To join the mailing list go to the website.

If you use a calendar system, you can see the individual seminars at bit.ly/zag-seminar-calendar

Times vary to accommodate speakers time zones but times will be announced in GMT time.

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