Speaker: Fabrizio Catanese (Universität Bayreuth)

Time: 16:00-17:00 p.m., Jun 3, 2025, GMT+8

Venue: Room 77201, Jingchunyuan 78, BICMR, PKU

Abstract: 

In the beginning, elementary, part I shall explain some basic notions about binary codes, vector spaces over the field with 2 elements, relying on basic linear algebra. I will then explain the relation between coding theory and families of singular surfaces X, based on the first homology of the nonsingular part X* of X and of the complement in X* of a hyperplane section D. If X is nodal, we get two binary codes K and K’. The following result answers a problem open since more than 150 years.

Theorem 1: (D^2=4) the set of components of the space of nodal quartic surfaces in P^3 with d nodes is in bijection with the set of isomorphism classes of d-binary codes K’which are the shortenings of the so called Kummer code. I will show other similar theorems for the case of all K3 surfaces of any degree D^2: everything can be done without computers, using coding and lattice theory. I will finally describe a new coding theory, associated to normal surfaces: this should help to treat similarly K3 surfaces with ADE singularities.

Source: Beijing International Center for Mathematical Research, PKU