Resolution of singularities of an idealistic filtration in dimension 3 after Benito-Villamayor

Abstract

We establish an algorithm for resolution of singularities of an idealistic filtration in dimension 3 (a local version) in positive characteristic, incorporating the method recently developed by Benito-Villamayor into our framework. Although (a global version of) our algorithm only implies embedded resolution of surfaces in a smooth ambient space of dimension 3, a classical result known before, we introduce some new invariant which effectively measures how much the singularities are improved in the process of our algorithm and which strictly decreases after each blow up. This is in contrast to the wellknown Abhyankar-Moh pathology of the increase of the residual order under blow up and the phenomenon of the "Kangaroo" points observed by Hauser.2010 Mathematics Subject Classification. 14E15.

A new strategy for resolution of singularities in the monomial case in positive characteristic

“…We remark that the problem of resolution of singularities of a basic object (W, (I, a), E) is reduced to the problem of resolution of singularities of an idealistic filtration (W, R, E) if we set R = n∈Z ≥0 (I ⌈ n a ⌉ , n). We also remark that what we actually discuss in this paper is the following local version of the above problem (as we discussed only the local version in our previous paper [22]): Starting from a closed point P ∈ Sing(R) ⊂ W and its neighborhood, we have a sequence of closed points and their neighborhoods P ∈ Sing(R) ⊂ W P 0 ∈ Sing(R 0 ) ⊂ W 0 ←− P 1 ∈ Sing(R 1 ) ⊂ W 1 ←− · · · ←− P i ∈ Sing(R i ) ⊂ W i in the resolution sequence. After we choose the center P i ∈ C i ⊂ Sing(R i ) ⊂ W i and take the corresponding transformation W i πi+1 ←− W i+1 to extend the resolution sequence, the "devil" tries to choose a closed point P i+1 ∈ π −1 i (P i ) ∩ Sing(R i+1 ) ⊂ W i .…”

“…type) is a finitely generated graded O W -algebra R = n∈Z ≥0 (I n , n), satisfying the condition O W = I 0 ⊃ I 1 ⊃ I 2 · · · ⊃ I n ⊃ · · · , where "n" in the second factor specifies the "level" of the ideal I n in the first factor, and that the singular locus of an idealistic filtration is defined to be Sing(R) := {P ∈ W | ord P (I n ) ≥ n, ∀n ∈ Z ≥0 }. For the precise definition of a transformation π i+1 , we refer the reader to [22].…”

“…(3) In our algorithm, one cannot expect in general to have the idealistic filtration R P in the monomial case to be D-saturated, D E -saturated, or D Eyoung -saturated, since we only take the pull-back of the idealistic filtration under transformation, without taking any further saturation, when the invariant σ stays the same (cf. [22]). It seems that the partial saturation as described in condition 4 in the setting is the best we can hope for in our current algorithm.…”

“…We summarize, in the table below, the procedure for resolution of singularities in the monomial case, according to the value of the invariant τ (cf. [22]).…”

“…(3) We refer the reader to 5.1 in [22] for a more detailed case analysis according to the value of the invariant τ in dimension d = 3.…”