Bernd Schober 
(Université de Versailles) 

Título "Characteristic Polyhedra of Singularities without
completion" 

Fecha y Lugar "05.06.2014, 12:30. 
Seminario A125 de la Facultad de Ciencias de la Universidad de Valladolid

 Resumen: 
Let $ R $ be a Noetherian regular local ring, $  (u ) = ( u_1, \ldots, u_e ) $ a system of regular elements in $ R $, and $ J \subset R $ a non-zero ideal.
For these data Hironaka introduced the characteristic polyhedron $ \Delta ( J;u) $ which yields crucial information on the singularity defined by $ J $.

In his original work Hironaka also gave a procedure how to compute $ \Delta ( J;u) $ starting with a given set of generators $ ( f ) = ( f_1, \ldots, f_m ) $ and a system of elements $ ( y) = ( y_1, \ldots, y_r ) $ which extends $ ( u ) $ to a regular system of parameters for $ R $.
But there exist examples where this process is not finite.
Recently V.~Cossart and O.~Piltant have shown that one can achieve $ \Delta ( J;u) $ with a different procedure in finitely many steps under the assumption that $ R $ is a $ G $-ring, $ J $ is a principal ideal ($ m = 1 $), and $ r = 1 $.
In this talk I show that their process is finite if $ J $ is any ideal, $ R $ is excellent, and the reduced ridge of $ J $ coincides with its directrix.

First, I recall Hironaka's definition of the characteristic polyhedron of a singularity.
In particular I explain the procedures of normalization and vertex solving with whom one can achieve the characteristic polyhedra in the completion of $ R $.
After that I show that the normalization process is always finite and then I reduce the proof for the finiteness of solving vertices to the case of an empty characteristic polyhedron.
All the previous steps are valid in any excellent regular local ring $ R $ without the assumption on the ridge.
Finally, I discuss the key case of an empty characteristic polyhedron, $ \Delta (J;u) = \emptyset $, and under which assumptions one can attain it without going to the completion.


(This is joint work with V.~Cossart)