Forced point centers and the residual-order cycle obstruction to resolution of singularities in positive characteristic

Hauser and Perlega (Journal of Algebraic Geometry 28, 2019) constructed, in every prime characteristic, purely inseparable hypersurface singularities that return to the same shape - with larger residual order - after a finite cycle of permissible point blowups, so that iteration drives the residual order to infinity while the order of the defining equation stays constant, disproving Moh's claimed stability theorem for the residual order. In their examples the singular locus is positive-dimensional, so larger permissible centers are always available and break the cycle; the construction therefore does not refute resolution of singularities in positive characteristic. The problem posed by the source is to decide whether this escape by larger centers is universal: construct a purely inseparable hypersurface singularity in positive characteristic for which the choice of point centers is forced (for instance because the singularity is isolated) and which admits such a non-terminating cycle - which the authors state would disprove the existence of resolution of singularities in positive characteristic - or prove that no such example exists, for example by exhibiting a terminating resolution procedure in arbitrary dimension and characteristic based on an invariant controlling both the main order and the residual data.

Background

Resolution of singularities asks whether every algebraic variety $X$ over a field $k$ admits a proper birational map $X'\to X$ with $X'$ regular. Hironaka (1964) proved this over fields of characteristic zero in the stronger embedded form: for $X$ embedded in a regular ambient scheme $W$, a finite sequence of blowups of $W$ along regular centers makes the strict transform of $X$ regular and transverse to the exceptional divisor. In positive characteristic the problem is open in general: non-embedded resolution is known up to dimension three (Cossart and Piltant, 2019, building on the classical surface case), and embedded resolution up to dimension two.

The characteristic-zero proof is inductive. One blows up the locus where the order $\operatorname{ord}_a X$ of a defining ideal is maximal; at points where this order stays constant under blowup, one descends to a hypersurface of maximal contact - a smooth hypersurface containing the worst singular locus - and uses the order there of the coefficient ideal (the ideal on the hypersurface encoding the transverse structure of the singularity), minus the multiplicities of the exceptional components, as the next component of a resolution invariant whose lexicographic decrease drives termination. In positive characteristic, hypersurfaces of maximal contact need not exist, and the substitute is to take the maximum of this number over all smooth local hypersurfaces: the residual order, in Hironaka's terminology. A permissible center is a regular center contained in the equimultiple locus of the defining ideal (the locus of points where the order of the ideal attains its maximum) and having normal crossings with the exceptional divisor. Two characteristic-specific pathologies then appear: the residual order is not upper semicontinuous, and it may increase under a permissible blowup at a point where the order of the defining ideal has remained constant - the 'kangaroo phenomenon' in Hauser's terminology (Hauser, 2010).

For purely inseparable hypersurface equations $f=z^{p^e}+F(x_1,\dots,x_n)$ over an algebraically closed field of characteristic $p>0$, Moh (1987) proved that a single permissible blowup increases the residual order by at most $p^{e-1}$, and claimed a Stability Theorem: along any sequence of permissible blowups under which the order of the strict transform of $f$ stays constant, the residual order cannot grow beyond $d+p^{e-1}$, where $d$ is its initial value. This would have ruled out indefinite growth and supported a resolution algorithm built on the pair (order, residual order).

Hauser and Perlega (2019) disproved this claim: for every prime $p$ they constructed purely inseparable hypersurface singularities (with $e=3$, i.e. order $p^3$) and finite cycles of permissible point blowups after which the defining equation returns to exactly the same shape with certain exponents - and the residual order - shifted upwards, so that iteration gives an infinite permissible sequence with constant order and residual order tending to infinity; they also located the flaw in Moh's argument, an unjustified divisibility reduction. Crucially, in all their examples the singular locus is positive-dimensional, so larger permissible centers are available at every stage and break the cycle: the examples are not counterexamples to resolution. In parallel work the same authors showed that the invariant program does work for surfaces (embedded resolution of surface singularities in arbitrary characteristic, Hauser and Perlega 2024), and that an increase of the residual order is a rare, explicitly characterizable event (Hauser and Perlega 2019). From dimension three upward, however, no efficient invariant for embedded resolution is known, and the general resolution problem in positive characteristic remains open.

Problem Statement

Decide whether the escape by larger permissible centers, available in every known cycle example, is universal - the question Hauser and Perlega pose in the Introduction of their paper.

Let $K$ be an algebraically closed field of characteristic $p>0$, and let $f=z^{p^e}+F(x_1,\dots,x_n)$ define a purely inseparable hypersurface singularity, with residual order taken with respect to the exceptional divisor and with permissible centers as defined in the Background.

(1) Either construct a cycle with forced point centers: a purely inseparable hypersurface singularity in positive characteristic for which the choice of point centers is forced - for instance because the singularity and its transforms along the sequence are isolated - together with an explicit sequence of permissible point blowups under which the order of the strict transform of $f$ remains constant while the residual order tends to infinity, with the forcedness of the point centers and the non-improvement of the residual order holding along all permissible choices of centers, not merely along the chosen sequence: a single divergent sequence while larger permissible centers remain available - as in the source's own examples - is explicitly not a counterexample to resolution. By the source paper's assertion, such an example would disprove the existence of resolution of singularities in positive characteristic.

(2) Or prove that such examples cannot exist: that no purely inseparable hypersurface singularity in positive characteristic admits forced-centers cycles with unbounded residual order. The strongest form of such an answer is the general termination question the source poses: a terminating resolution procedure in arbitrary dimension and characteristic, using permissible (possibly higher-dimensional) centers and an invariant controlling both the main order and the residual data - that is, embedded resolution of singularities in positive characteristic in arbitrary dimension, the famous open problem of which this is the source's proposed route.

References

1. Herwig Hauser and Stefan Perlega. Cycles of singularities appearing in the resolution problem in positive characteristic. Journal of Algebraic Geometry 28 (2019), 391-403. DOI: 10.1090/jag/718. https://doi.org/10.1090/jag/718 (preprint arXiv:1802.05010, https://arxiv.org/abs/1802.05010)
2. Herwig Hauser and Stefan Perlega. Characterizing the increase of the residual order under blowup in positive characteristic. Publications of the Research Institute for Mathematical Sciences 55 (2019), 835-857. DOI: 10.4171/prims/55-4-7. https://doi.org/10.4171/prims/55-4-7 (preprint arXiv:1906.09593, https://arxiv.org/abs/1906.09593)
3. Herwig Hauser and Stefan Perlega. Resolving surface singularities in positive characteristic. Publications of the Research Institute for Mathematical Sciences 60 (2024), 767-813. DOI: 10.4171/prims/60-4-5. https://doi.org/10.4171/prims/60-4-5
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