Partial desingularization in characteristic 0

Abstract: It is shown by Belotto da Silva and Bierstone [arXiv:2602.09114] and Włodarczyk [arxiv:2602.14266] that, if one allows to introduce stack theoretic weighted blowups, any variety $X$ over a field of characteristic 0 admits a normal crossings resolution. We introduce a principle that makes such results possible and inevitable, see Theorem 3.

• The paper establishes a general principle showing that a wonderful invariant and an adequate singularity class produce a modification that preserves the class and is unchanged over its locus.
• The authors verify adequacy for hypersurface normal crossings singularities using a normal-cone argument, homogeneity of invariants, and properness of the dual projective space.
• The framework clarifies recent characteristic-zero normal-crossings resolution results while identifying open problems for broader non-reduced, logarithmic, and foliated settings.

Context and main result

This note by Dan Abramovich and Michael Temkin addresses the problem of resolving singularities of a variety $X$ over a field of characteristic 0 only up to its normal crossings locus — that is, producing a modification $X' \to X$ where $X'$ has only normal crossings (nc) singularities and which is an isomorphism on the nc locus of $X$. Such a map is called a normal crossings resolution. The motivating result is due independently to Belotto da Silva–Bierstone and to Włodarczyk: any pure-dimensional variety in characteristic 0 admits a functorial stack-theoretic normal-crossings resolution, provided one allows Deligne--Mumford stacks arising from weighted blowups (Silva et al., 9 Feb 2026, Włodarczyk, 15 Feb 2026). This resolves a question posed by Kollár (Problem 9 of "Semi log resolution" (0812.3592)) at the cost of allowing abelian stabilizers; passing to coarse moduli spaces, one obtains resolutions whose target has singularities that are abelian permutation-quotients of normal crossings singularities — a class strictly wider than the hypersurface singularities obtained by Belotto da Silva and Bierstone.

The paper's contribution is not a new resolution algorithm but an extraction of the structural principle underlying these results. The authors state explicitly that they were working on Whitney umbrellas in characteristic 0 and 2 when they found the argument; upon sharing it, they learned of the concurrent papers of Belotto da Silva–Bierstone and Włodarczyk, and repurposed their manuscript to isolate the principle.

Wonderful invariants and adequate classes

The framework rests on two notions. A wonderful invariant is a function $\iota\colon |X| \to \Gamma$, valued in a well-ordered set, satisfying four properties drawn from the invariant of Abramovich–Temkin–Włodarczyk for weighted blowups:

• $\iota$ is upper semicontinuous and takes its minimum precisely on smooth points;
• $\iota$ is functorial for smooth morphisms;
• level sets $W_K = \{p : \iota_X(p) = K\}$ are smooth;
• level sets are immediately reducible: $W_K$ carries a weighted blowup center $J_K$ such that blowing up $X' \to X$0 drops the maximal invariant below $X' \to X$1.

An adequate class $X' \to X$2 of singularities satisfies two conditions: it has open support (the locus of points with singularity in $X' \to X$3 is open), and it is $X' \to X$4-closed: within each level set $X' \to X$5, the locus where $X' \to X$6 and the point lies in $X' \to X$7 is closed. For the nc context, the invariant of an order-$X' \to X$8 nc point ($X' \to X$9-point) is $X'$0 with $X'$1 entries.

The principle and its proof

The main theorem asserts: given a wonderful invariant $X'$2 and an adequate class $X'$3, every variety $X'$4 admits a modification $X'$5 such that $X'$6 has only $X'$7-singularities and $X'$8 is an isomorphism over the locus of $X'$9 points. The proof is a straightforward induction on the well-ordered invariant value $X$0. At the top stratum, upper semicontinuity makes $X$1 closed, adequacy makes $X$2 closed in $X$3 and hence closed in $X$4 while open support makes $X$5 open. One removes a neighborhood of $X$6, applies the immediately reducing blowup — an isomorphism over the remaining part of the $X$7 locus — and then applies the induction hypothesis stack-theoretically. Gluing the resulting modification with the identity on the neighborhood of $X$8 yields $X$9. The argument uses classical resolution methods to reduce first to the case where $\iota\colon |X| \to \Gamma$0 is a hypersurface divisor in a smooth $\iota\colon |X| \to \Gamma$1; Włodarczyk's version avoids this step at the price of discussing higher-codimension invariants.

Closedness of the nc locus

The key verification that the class of hypersurface normal crossings singularities is adequate reduces to a lemma: for $\iota\colon |X| \to \Gamma$2 a hypersurface in characteristic 0 and $\iota\colon |X| \to \Gamma$3, the locus $\iota\colon |X| \to \Gamma$4 of $\iota\colon |X| \to \Gamma$5-points is closed in the level set $\iota\colon |X| \to \Gamma$6. The proof proceeds by reduction to the normal cone of $\iota\colon |X| \to \Gamma$7 in $\iota\colon |X| \to \Gamma$8: since the invariant is homogeneous of weight $\iota\colon |X| \to \Gamma$9 along $\iota$0, replacing $\iota$1 by the normal cone preserves the invariant $\iota$2, and the normal cone of an nc point is again nc. Conversely, if the fiber of the cone over a geometric point $\iota$3 has ideal generated by a product of $\iota$4 linearly independent linear forms lifting to divisors of the defining equation near the generic point of $\iota$5, étale descent gives an nc form near $\iota$6. The fiber itself is the vanishing locus of a homogeneous degree-$\iota$7 form in affine $\iota$8-space; properness of $\iota$9 supplies limiting linear forms from the closure of $\iota$0, and linear independence follows because dependence would force the invariant of the fiber to be $\iota$1 with fewer than $\iota$2 terms, i.e. $\iota$3. A concrete exclusion is noted: the Whitney umbrella $\iota$4 cannot appear in $\iota$5, since along the nc locus the invariant is $\iota$6 while the normal cone's invariant at the origin would be $\iota$7, and the umbrella's origin has invariant $\iota$8.

With this lemma, Theorem (nc) follows directly from the general principle. As a corollary of the same mechanism, Włodarczyk's stronger theorem for non-reduced nc singularities, étale locally of the form $\iota$9 for local parameters $W_K = \{p : \iota_X(p) = K\}$0, also fits the framework: the authors state that the argument of the lemma applies verbatim in that setting.

Limitations and scope

The paper is candid about the reach of the method. The adequate class $W_K = \{p : \iota_X(p) = K\}$1 is trivially adequate but yields nothing new. The authors observe that applying the principle to nc singularities with controlled logarithmic exceptional divisors, using the logarithmic invariants of Włodarczyk and Abramovich–Belotto da Silva–Quek–Temkin–Włodarczyk, "would likely not go beyond" the existing results. Two genuine open directions are flagged: whether the principle extends to non-reduced normal crossings classes beyond what Włodarczyk already proves, and — left entirely unaddressed — the foliated case treated in the principalization work of Abramovich–Belotto da Silva–Temkin–Włodarczyk (Abramovich et al., 2 Mar 2025). The material is expected to be folded back into the authors' manuscript on Whitney umbrellas in characteristics 0 and 2 when completed.

Conclusion

The paper distills the simultaneous results of Belotto da Silva–Bierstone and Włodarczyk into a short, reusable induction: any wonderful invariant together with an adequate (open-support, invariant-closed) class of singularities yields a partial desingularization preserving that class. The substantive input beyond bookkeeping is the normal-cone argument establishing closedness of the order-$W_K = \{p : \iota_X(p) = K\}$2 nc locus within its invariant level set, which relies on properness of the dual projective space of linear forms and on the homogeneity of the invariant. Whether analogous adequate classes exist in the logarithmic-foliated setting remains the natural open question raised by the note.