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Seminar on Algebraic Geometry

Tuesday, 8.11. at 10:15
ricam seminar room (HF 136)

Oliver Labs
Hypersurfaces with Many Singularities.

One of the first questions that geometers asked about algebraic hypersurfaces was which and how many singularities can occur on them. It turned out that this is a very difficult problem in general which is only solved in a few cases.

In three-space, Schlaefli's classification of cubic surfaces w.r.t. their singularities is classical (1863). But already for quartic surfaces all possible configurations of singularities on them where only found recently (1997). For surfaces of higher degree not much is known apart from upper and lower bounds for the maximum possible number of singularities that can occur.

We briefly review the the current state of knowledge. Then we present the results obtained in our Ph.D. thesis. These mainly consist of constructions which lead to new lower bounds. Some of our methods are purely theoretical in nature, others use computer algebra.

A nice feature of the subject is that it is often possible to give beautiful visualizations of the hypersurfaces. During the talk, we will show how to use our software surfex for these purposes.