2010/04/11 by Benito, Angélica, Villamayor, Orlando E.
#Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1004.1803
The problem of resolution of singularities in positive characteristic can be reformulated as follows: Fix a hypersurface X, embedded in a smooth scheme, with points of multiplicity at most n. Let an n-sequence of transformations of X be a finite composition of monoidal transformations with centers included in the n-fold points of X, and of its successive strict transforms. The open problem (in positive characteristic) is to prove that there is an n-sequence such that the final strict transform of X has no points of multiplicity n (no n-fold points). In characteristic zero, such an n-sequence is defined in two steps: the first consisting in the transformation of X to a hypersurface with n-fold points in the so called monomial case. The second step consists in the elimination of these n-fold points (in the monomial case), which is achieved by a simple combinatorial procedure for choices of centers. The invariants treated in this work allow us to define a notion of strong monomial case which parallels that of monomial case in characteristic zero: If a hypersurface is within the strong monomial case we prove that a resolution can be achieved in a combinatorial manner.