Hironaka's Desingularization

Updated 22 November 2025

Hironaka’s desingularization is a process that transforms singular algebraic varieties into nonsingular ones using a finite sequence of blow-ups along smooth centers.
The method employs resolution invariants and marked ideals to guide algorithmic, functorial operations that ensure canonical and effective results.
Modern refinements integrate combinatorial tools like the characteristic polyhedron and arc theoretic interpretations to enhance the process and computational efficiency.

Hironaka's desingularization, or resolution of singularities, refers to the process of transforming a singular algebraic variety into a nonsingular one by a finite sequence of explicit geometric operations—most classically, blow-ups along carefully chosen smooth centers. The foundation of this theory, established by Hironaka in 1964, guarantees the existence of resolutions in characteristic zero. Since then, the theory has evolved into highly algorithmic, functorial, and effective frameworks, with deep connections to valuation theory, arc spaces, and combinatorial invariants such as Hironaka's characteristic polyhedron.

1. Formalism and Invariants

Let $X$ be a smooth algebraic variety over a field of characteristic zero, and let $\mathcal{I} \subset \mathcal{O}_X$ be a coherent ideal sheaf. The order function is defined by

$\ord_x(\mathcal{I}) := \max\{k \ge 0 \mid \mathcal{I}_x \subset \mathfrak{m}_x^k\}$

for any $x \in X$ , recovering the standard notion of vanishing order for principal ideals. The primary object in modern approaches is the marked ideal $(X, \mathcal{I}, E, p)$ , consisting of a smooth variety, a coherent ideal, an ordered simple normal-crossings divisor $E$ , and an integer $p > 0$ . The singular locus or support is

$\Supp(\mathcal{I}, p) = \{x \in X \mid \ord_x(\mathcal{I}) \geq p\}.$

Under a blow-up along a permissible smooth center $C \subset \Supp(\mathcal{I}, p)$ transverse to $E$ , the controlled transform is given by

$\mathcal{I}' = \mathcal{O}_{X'}(-p D) \cdot \sigma^* (\mathcal{I}),$

where $D$ is the exceptional divisor and $E' = \sigma_*^{-1}(E) + D$ .

The key resolution invariants are sequences of rational numbers and combinatorial data (orders, residual multiplicities, divisor counts), assembled into tuples such as

$\operatorname{inv}_X(a) = (v_1(a), s_1(a); v_2(a), s_2(a); \dots)$

where $v_1(a) = \ord_a \mathcal{I}$, $s_1(a)$ counts exceptional divisors, and higher $v_k, s_k$ are computed inductively via maximal contact and coefficient ideals (Bierstone et al., 2011).

2. Hironaka's Theorems and Algorithmic Schemes

Hironaka's embedded resolution theorem asserts that for any closed subvariety $Y \subset X$ of a smooth variety (char 0), there exists a canonical sequence of blow-ups of smooth centers such that the final strict transform of $Y$ is smooth and meets the total exceptional divisor only with simple normal crossings; the construction is functorial for smooth morphisms and equivariant under group actions (Bierstone et al., 2012, 1711.09976).

The principalization theorem states that every ideal admits a canonical sequence of blow-ups resulting in a simple normal crossings divisor (Bierstone et al., 2012).

Modern refinements yield a fully algorithmic approach: at each step, one considers the maximal locus of the resolution invariant and blows up that smooth stratum. Iteratively, this process lowers the invariant lexicographically, ensuring well-founded termination (Bierstone et al., 2011, Benito et al., 2011).

3. Constructive and Functorial Methods

The constructive algorithms rely on lexicographically ordered invariants derived from Rees algebras, maximal contact hypersurfaces, coefficient ideals, and their canonical homogenized forms to ensure functoriality and independence from arbitrary choices (Benito et al., 2011, Bierstone et al., 2012, Wlodarczyk, 2015). Explicitly, the stepwise reduction proceeds via:

Order reduction: At each maximal center where the order is maximal, replace the problem by resolving the coefficient ideal on a hypersurface of maximal contact.
Inductive dimension reduction: This leverages the invariance and compatibility of coefficient ideals with blow-ups and under étale/smooth morphisms.
Stratifications: The singular locus is stratified by the invariant, ensuring that each blow-up only modifies the worst singularities.

Functoriality with respect to group actions, smooth morphisms, étale localization, field extensions, and independence of embedding are all established via the invariance properties of these constructions (Benito et al., 2011, Wlodarczyk, 2015).

4. Complexity and Effectivity

Explicit complexity bounds for Hironaka's algorithm have been given: if $X$ is an affine $n$ -fold defined by polynomials of degree $\leq d_0$ and $\mathcal{I}$ is generated by $\ell$ polynomials, the number of blow-ups, maximal degree of transforms, and relevant complexities are all bounded by a primitive-recursive function in the Grzegorczyk class $E^{m+3}$ , with blow-up degrees typically increasing double-exponentially in $n$ at each step (Bierstone et al., 2012). In particular, the total bit-complexity is explicitly controlled, though only within a very coarse bound.

The method is constructive and canonical: all choices—such as maximal centers and maximal-contact hypersurfaces—are determined via invariants and do not depend on ad hoc data, thus ensuring uniqueness and functoriality (Bierstone et al., 2012, Wlodarczyk, 2015, 1711.09976).

5. Characteristic Polyhedron and Choice of Centers

Hironaka's characteristic polyhedron $\Delta$ is an essential combinatorial object controlling the choice of centers for blow-ups (Cossart et al., 2018, Abramovich et al., 1 Jul 2025). Given a well-prepared local expansion

$f = y^\nu + \sum_{(A,b)} c_{A,b} x^A y^b$

for $f$ in a regular local ring, the projected polyhedron is

$\Delta(f; x; y) = \mathrm{Conv}\left\{\frac{A}{\nu-b} + \mathbb{R}_{\geq 0}\ |\ c_{A,b}\ne 0 \right\}.$

Key numerical invariants, such as the minimal value $\delta$ in $\Delta$ , guide the selection of weighted or monomial blow-up centers.

The combinatorial face structure of $\Delta$ is invariant under changes of embedding and reflects deep properties of the singularity (e.g., multiplicity, Hilbert–Samuel function), ensuring that locally defined resolution steps globalize consistently (Cossart et al., 2018, Abramovich et al., 1 Jul 2025). In recent developments, weighted blow-ups guided by the data from $\Delta$ provide effective and sometimes faster functorial resolution algorithms, especially in dimension two and for plane curves (McQuillan et al., 2019, Abramovich et al., 1 Jul 2025).

6. Arc Theoretic and Valuative Interpretations

Hironaka's order function admits an intrinsic interpretation in terms of arcs and contact loci: the order at a singular point can be recovered as the asymptotic minimal normalized contact order (Nash multiplicity sequence) over all arcs centered at that point (Bravo et al., 2018, Bravo et al., 2018). The generic behavior of arcs under successive blow-ups encodes the geometry of desingularization and links the resolution algorithm with valuation theory and motivic integration. This perspective reveals the resolution invariant as the minimal rate at which the multiplicity of $X$ drops along arc-generated blow-up sequences.

In particular, the order invariant coincides with minima over the space of normalized contact orders, and the faces of the characteristic polyhedron correspond to essential valuations of the singularity (Bravo et al., 2018, Cossart et al., 2018).

Modern algorithms, such as those of Bierstone–Milman and Włodarczyk, introduce additional normalization—via cleaning steps and finer invariants—to avoid unnecessary blow-ups (e.g., preserving simple normal crossings loci) and minimize the class of unavoidable singularities after resolution (Bierstone et al., 2011, Bierstone et al., 2012, Włodarczyk, 2020).

Strong resolution with normally flat centers replaces classical maximal contact and principalization with functorial, Samuel-stratum-based approaches using canonical Rees algebras and standard bases, yielding a strengthened (and often shorter) desingularization process in characteristic zero (Wlodarczyk, 2015).

Functorial and canonical algorithms have been developed for singularities on orbifolds, toroidal, and toric spaces via explicit combinatorial and stack-theoretic constructions, unifying the classical and modern approaches (Włodarczyk, 2020, McQuillan et al., 2019).

Key references: (Bierstone et al., 2012, Bierstone et al., 2011, Benito et al., 2011, Cossart et al., 2018, Bravo et al., 2018, Bravo et al., 2018, 1711.09976, Wlodarczyk, 2015, McQuillan et al., 2019, Włodarczyk, 2020, Abramovich et al., 1 Jul 2025)