Gluing local resolutions into a global resolution over perfect fields, and geometric resolution of singular moduli spaces (open problems from Hu's 'Grassmannians and Singularities')

The preprint 'Grassmannians and Singularities' (Y. Hu, arXiv:2109.02968v4, 2022) proves that every singular closed point of an integral scheme of finite presentation over a perfect field has an open neighborhood admitting a resolution, via explicit blowups of Grassmannian strata combined with Mnëv universality. The author leaves three tasks open: (a) glue the finitely many local resolutions into a single global resolution — the new content being positive characteristic, where global resolution is a famous open problem; (b) Problem 11.1, a geometric alternative: resolve any singular moduli space by enlarging the moduli problem with boundary data that reduces automorphisms, so that all reduced irreducible components become smooth and meet transversally; (c) an acknowledged practical weakness, the inefficiency of the three-stage blowup process, for which the source poses no sharp criterion. The author's later work resolves only the genus-one and genus-two instances of (b) (references 10–12).

Background

A resolution of an integral (irreducible and reduced) algebraic scheme $X$ is a smooth scheme $\widetilde{X}$ together with a proper birational morphism $\widetilde{X}\to X$; 'birational' means the morphism is an isomorphism over a dense open subset, i.e. it changes $X$ only along a lower-dimensional locus. A perfect field is one whose field extensions are all separable; finite fields and fields of characteristic zero are perfect. Hironaka's 1964 theorem establishes resolution for all varieties over fields of characteristic zero, globally. In positive characteristic, resolution is known for surfaces and for threefolds in any characteristic (Cossart–Piltant, building on earlier work), and remains open in dimension at least four; the best general statement in arbitrary characteristic is de Jong's theorem of alterations (1996), which gives a proper, surjective, generically finite — but not birational — cover by a smooth variety. Historically, Zariski's 1940 approach proved local statements first (local uniformization along valuations) and then could not complete the patching step in general; the difficulty of gluing local birational pieces into one global smooth model is a recognized obstacle, since natural gluing operations such as fiber products and graph closures need not preserve smoothness.

The preprint (arXiv:2109.02968, 148 pages, v4 of March 2022; unpublished) takes a new route. By Mnëv's universality theorem — every integral singularity of finite type over $\mathbb{Z}$ appears in a configuration space of points, or scheme-theoretically (Lafforgue) in a 'thin Schubert cell' of the Grassmannian $\mathrm{Gr}_{3,E}$ of 3-dimensional subspaces of a vector space $E$, i.e. an intersection of codimension-one Schubert cells, described in Plücker coordinates on the exterior power $\bigwedge^3 E$ — it suffices to resolve all integral closed subschemes (Γ-schemes) of affine charts of $\mathrm{Gr}_{3,E}$. The paper constructs, by an explicit three-stage sequence of blowups (denoted $\vartheta$-, $\wp$-, and $\mathfrak{d}$-blowups, inductively transforming the defining Plücker-type equations and analyzing their Jacobians), a smooth 'transform' dominating each integral Γ-scheme over $\mathbb{Q}$ and over finite fields, and then transfers this to arbitrary perfect base fields by spreading-out — a standard specialization technique that carries statements proved over a prime field to its extension fields — from the prime subfield. The resulting Theorem 1.1 is local: for each singular closed point $q$ of an integral scheme $X$ of finite presentation over a perfect field, some open neighborhood $V\ni q$ admits a resolution $\widetilde{V}\to V$. The author states that gluing these local resolutions into a global one 'is being pursued', invoking the quasi-compactness of the Riemann–Zariski space of $X$ — the set of valuation rings of the function field $K(X)$ containing the base field, endowed with the Zariski topology — to reduce to finitely many patches. Because characteristic zero is covered by Hironaka, the substantive content of the gluing task is positive characteristic, where a global resolution theorem in dimension at least four would be a major breakthrough.

The preprint also proposes a different, non-algorithmic route for moduli spaces. A moduli space is an algebraic scheme or stack whose points classify geometric objects (curves, maps, sheaves, etc.); singularities of moduli spaces typically arise where classified objects have automorphisms or obstructed deformations. Problem 11.1 of the preprint asks to resolve such singularities conceptually: instead of blowing up equations, modify the moduli problem itself — adding extra data to the boundary objects (the degenerate objects parametrized by the boundary) so that automorphisms are reduced and removable obstructions eliminated — and obtain a new moduli space whose reduced irreducible components are smooth and meet transversally (i.e. cross like coordinate hyperplanes). The preprint's own blowup procedure is acknowledged to be 'highly inefficient' as a practical matter, though no optimality criterion is specified. The author's subsequent peer-reviewed work implements this modular-resolution philosophy for genus-one curves ('twisted fields' — an added-data construction that kills boundary automorphisms — with J. Niu, 2021) and for genus-two stable maps (maps from nodal curves of genus two to a projective target, modulo equivalence, with finite automorphism groups; with J. Li and J. Niu, 2026), but the general problem for arbitrary singular moduli spaces is untouched.

Problem Statement

The problem is to complete the resolution program of the preprint 'Grassmannians and Singularities' (arXiv:2109.02968) over perfect fields, along its two stated precise open directions, which share one scientific objective — establishing resolution of singularities for integral schemes over perfect fields, algorithmically or geometrically. The two directions are independent open tasks: completing either one constitutes a full resolution of that direction, and neither is a prerequisite for the other.

(a) Global gluing. Let $X$ be an integral singular scheme of finite presentation over a perfect field $k$. The preprint's Theorem 1.1 provides, for each singular closed point $q\in X$, an open neighborhood $V\ni q$ and a resolution $\widetilde{V}\to V$ (a smooth $k$-scheme mapping properly and birationally onto $V$). Construct from these local data a single resolution $\widetilde{X}\to X$ of the whole scheme: $\widetilde{X}$ smooth over $k$, the morphism proper and birational, and the construction valid at every point of $X$. Equivalently, by quasi-compactness, glue finitely many of the constructed local resolutions into one global smooth model. In characteristic zero this is Hironaka's theorem, so the substantive requirement is that the construction be valid over perfect fields of positive characteristic (any dimension).

(b) Geometric resolution of moduli (Problem 11.1 of the preprint). For a singular moduli space $\mathfrak{M}$, find a moduli space $\widetilde{\mathfrak{M}}$ that modifies only the boundary objects of $\mathfrak{M}$ — each modified object being obtained from an original one by adding extra data that reduces its automorphism group and, ideally, removes all removable obstructions — such that every irreducible component of $\widetilde{\mathfrak{M}}$, endowed with its reduced structure, is smooth, and distinct components meet transversally (see Conjectures 5.4 and 5.5 of [13] for the precise formulation of the boundary condition).

As auxiliary context (not a sharply posed subproblem): the preprint acknowledges that its explicit three-stage $\vartheta$-, $\wp$-, $\mathfrak{d}$-blowup procedure is highly inefficient in practice, and improving or simplifying such blowup sequences while preserving the resolution property is left open; the source specifies no optimality criterion or completion boundary for this item.

References

1. Yi Hu, Grassmannians and Singularities, preprint arXiv:2109.02968 (v4, March 2022, 148 pp.). DOI: 10.48550/arXiv.2109.02968, https://arxiv.org/abs/2109.02968
2. Heisuke Hironaka, Resolution of Singularities of an Algebraic Variety Over a Field of Characteristic Zero: I, Annals of Mathematics (2) 79 (1964), no. 1, 109–203. DOI: 10.2307/1970486, https://doi.org/10.2307/1970486
3. Heisuke Hironaka, Resolution of Singularities of an Algebraic Variety Over a Field of Characteristic Zero: II, Annals of Mathematics (2) 79 (1964), no. 2, 205–326. DOI: 10.2307/1970547, https://doi.org/10.2307/1970547
4. A. J. de Jong, Smoothness, semi-stability and alterations, Publications Mathématiques de l'IHÉS 83 (1996), 51–93. DOI: 10.1007/bf02698644, https://doi.org/10.1007/bf02698644
5. Vincent Cossart and Olivier Piltant, Resolution of singularities of threefolds in positive characteristic. I, Journal of Algebra 320 (2008), no. 3, 1051–1082. DOI: 10.1016/j.jalgebra.2008.03.032, https://doi.org/10.1016/j.jalgebra.2008.03.032
6. Vincent Cossart and Olivier Piltant, Resolution of singularities of threefolds in positive characteristic. II, Journal of Algebra 321 (2009), no. 7, 1836–1976. DOI: 10.1016/j.jalgebra.2008.11.030, https://doi.org/10.1016/j.jalgebra.2008.11.030
7. Oscar Zariski, Local Uniformization on Algebraic Varieties, Annals of Mathematics (2) 41 (1940), no. 4, 852–896. DOI: 10.2307/1968864, https://doi.org/10.2307/1968864
8. N. E. Mnëv, The universality theorems on the classification problem of configuration varieties and convex polytopes varieties, in: Topology and Geometry — Rohlin Seminar, Lecture Notes in Mathematics 1346, Springer, Berlin (1988), 527–543. DOI: 10.1007/bfb0082792, https://doi.org/10.1007/bfb0082792
9. L. Lafforgue, Pavages des simplexes, schémas de graphes recollés et compactification des $PGL_{r}^{n+1}/PGL_{r}^{r}$, Inventiones Mathematicae 136 (1999), no. 1, 233–271. DOI: 10.1007/s002220050309, https://doi.org/10.1007/s002220050309
10. Yi Hu and Jingchen Niu, Moduli of curves of genus one with twisted fields, Asian Journal of Mathematics 25 (2021), no. 5, 683–714. DOI: 10.4310/ajm.2021.v25.n5.a4, https://doi.org/10.4310/ajm.2021.v25.n5.a4
11. Yi Hu and Jingchen Niu, A theory of stacks with twisted fields and resolution of moduli of genus two stable maps, Asian Journal of Mathematics 29 (2025), no. 3, 325–422. DOI: 10.4310/ajm.250905113140, https://doi.org/10.4310/ajm.250905113140
12. Yi Hu, Jun Li, and Jingchen Niu, Genus two stable maps, local equations and modular resolutions, Moduli 3 (2026). DOI: 10.1017/s2949764726100265, https://doi.org/10.1017/s2949764726100265
13. Yi Hu, Relative Resolution and Its Applications, in: Proceedings of the International Congress of Chinese Mathematicians (ICCM), 2017; preprint arXiv:1310.2237. DOI: 10.48550/arXiv.1310.2237, https://arxiv.org/abs/1310.2237