Resolution of singularities for 3-folds in positive characteristic

2009 ◽  
Vol 131 (1) ◽  
pp. 59-127 ◽  
Author(s):  
Steven Dale Cutkosky
Keyword(s):  
Positive Characteristic ◽  
Resolution Of Singularities

scholarly journals Monoidal transforms and invariants of singularities in positive characteristic

2013 ◽  
Vol 149 (8) ◽  
pp. 1267-1311 ◽  
Author(s):  
Angélica Benito ◽  
Orlando E. Villamayor U.
Keyword(s):  
Open Problem ◽  
Characteristic Zero ◽  
Positive Characteristic ◽  
Resolution Of Singularities ◽  
Second Step

AbstractThe 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 consists of the transformation of $X$ to a hypersurface with $n$-fold points in the so-called monomial case. The second step consists of 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 present 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.

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