(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
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)