Resolution of singularities in positive characteristic: the case of small dimension or large characteristic

• Applicant: Dr. Bernd Schober
• Subject Area: Mathematics
• Term: from 2013 to 2014
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 250489866

In algebraic geometry one studies the properties of geometric objects X. In the resolution of singularities one is especially interested if it is always possible to resolve knots, cusps and similar irregularities (called singularities). More precisely, one is looking for an object Y such that Y has no singularities, Y coincides with X almost everywhere and additionally X is a suitable projection (or more metaphorical a shadow) of Y.For objects X, which can be imagined in the real world, i.e. in dimension one (curves) and two (surfaces), it is known that such a Y always exists. Under appropriate restrictions this is even true in all dimensions. But the arbitrary case is still open.The previous restrictions don't give a sharp definition of those X for which Y can be constructed with the known methods. Thus the aim of sub-project (1) is to find sharp conditions for this.In dimension at most two there exists the approach to obtain Y by using the geometric structure of the singularities. In some special case Hironaka has been able to construct an invariant by using so called characteristic polyhedra. This invariant measures the singularities, improves in every step and after finitely many step one gets Y. The characteristic polyhedron is another geometric object which can be associated to X and which reflects the nature of the singularities up to some point. The general proof for the above approach in dimension at most two is indirect. Hence the idea of sub-project (2) is to study Hironaka's polyhedra in order to find a suitable invariant which works for the arbitrary case (in dimension at most two).In the computation of the characteristic polyhedra one needs to apply a certain procedure. For some special case it is known that this process is finite. The aim of sub-project (3) is to use and to extend the methods of the proof in the special case in order to prove the general case.In sub-project (4) those X are considered where the known methods fail to work. This problem is more than 25 years old. Therefore sub-project (4) is speculative. The first unknown case is in dimension three. Here the easiest case has already been solved by Cossart and Piltant. Despite the deepness of this problem I see an approach to generalize their idea to the next open case and the hope is to make first steps towards the solution of this problem.

• DFG Programme: Research Fellowships
• International Connection: France
• Host: Professor Dr. Vincent Cossart