Simple Normal Crossings Divisor

• Simple Normal Crossings Divisor is a reduced divisor whose local equations are given by coordinate functions, ensuring smooth components intersect transversely.
• They play a key role in resolution of singularities by enabling algorithmic blow-ups that preserve the normal crossings structure and guarantee functoriality.
• Their applications extend to logarithmic geometry and mirror symmetry, where they model degenerations, define log structures, and support effective resolution techniques.

A simple normal crossings (snc) divisor is a central object in birational geometry, resolution of singularities, and logarithmic algebraic geometry, where explicit control and functoriality of singularities and their resolutions are required. An snc divisor is a reduced divisor with a prescribed local analytic structure: in the neighborhood of any point, it is defined by the vanishing of a subset of coordinate functions, ensuring that its irreducible components are smooth and intersect transversely. This geometric simplicity makes snc divisors the foundational configuration for studying and resolving singularities, constructing log structures, and formulating effective and functorial resolution algorithms.

1. Local and Global Definition

An snc divisor $E$ on a smooth algebraic variety $Z$ over an algebraically closed field of characteristic zero is a reduced divisor such that, for every point $p \in Z$, there exist local parameters $t_1,\ldots,t_n \in \mathcal{O}_{Z,p}$ for which

$E = V(t_1 \cdots t_d),$

for some $0 \leq d \leq n$. Here, each $V(t_j)$ defines a smooth component passing through $p$, and these local equations guarantee that the components meet transversely. Globally, $E = \bigcup_i E_i$ is a union of smooth hypersurfaces $\{ E_i \}$, each $E_i$ is smooth, and for any intersection $E_{i_1} \cap \cdots \cap E_{i_r}$ (with $r \leq n$), the intersection is smooth of codimension $r$ if nonempty.

This structure makes snc divisors a combinatorial analog of toroidal embeddings; locally they model toric divisors defined by monomial equations in affine spaces.

2. Role in Resolution of Singularities

In modern desingularization theory, particularly after Hironaka, snc divisors serve as the "boundary" for both embedded and logarithmic resolutions. The goal is to resolve singularities of a given variety $X$ and/or a divisor $D$ so that their strict and exceptional transforms form an snc divisor in the ambient space, preserving explicit control over the intersection behavior at all stages.

Let $X\subset Y$ be a closed subvariety of a smooth $Y$, with $D\subset Y$ an snc divisor. A logarithmic resolution of the pair $(X,D)$ is a proper birational map $\pi:Y'\to Y$ such that:

• $Y'$ is smooth;
• the total transform $D' := \pi^*D + \operatorname{Exc}(\pi)$ is an snc divisor;
• the strict transform $X'$ is smooth and meets $D'$ transversely.

The procedure involves a finite sequence of blow-ups at smooth centers which, at every step, are chosen to have normal crossings with the current total transform of $E$, maintaining the snc property inductively (Bierstone et al., 10 Nov 2025, Abramovich et al., 17 Mar 2025).

3. Algorithmic and Functorial Aspects

The construction of log resolutions with snc divisors is achieved via algorithmic processes relying on marked ideals and invariants. The marked ideal formalism encodes the geometry by a quintuple $(Z, X, E, \mathcal J, \mu)$, where $E$ is an ordered snc divisor on the smooth ambient variety $Z$.

The resolution process proceeds by repeatedly blowing up admissible centers having normal crossings with the current total transform of $E$. Centers are locally defined as loci where an explicit upper semicontinuous invariant $$\operatorname{inv}(p) = (\nu_1, s_1, ..., \nu_r, s_r, \nu_{r+1})$$ is maximal; the snc property is preserved at each stage. The process is functorial with respect to smooth morphisms and compatible with group actions and restrictions to open subsets, ensuring that local choices glue to global blow-ups in the projective embedding (Bierstone et al., 10 Nov 2025, Abramovich et al., 17 Mar 2025, Temkin, 2023).

The following table summarizes the transformation process:

Step	Structure Preserved	Centers
Initial $(X, E)$	$E$ is snc	—
After each blowing-up	$E'$ is snc	Smooth, nc with existing $E_j$
Final $(X', E')$	$E'$ is snc, $X'$ smooth	Smooth, intersection is snc

Super-exponential degree and dimension growth occur in general; the recurrences for the post-resolution embedding dimension $n'(n,d)$ and degree $d'(n,d)$ are of tower type, but this complexity is controlled via explicit bounds (Bierstone et al., 10 Nov 2025).

4. SNC Divisors in Logarithmic Geometry

snc divisors are the primary local models for divisorial log structures in logarithmic geometry. Given a smooth $X$ and snc divisor $D$, the "divisorial" log structure is $$M_X = \{ f \in \mathcal{O}_X : f|_{X \setminus D} \in \mathcal{O}_X^\times \}$$. Log regular log structures require (étale locally) that the model divisor be snc; log smoothness reduces locally to toroidal (i.e., snc) geometry (Temkin, 2023).

In the logarithmic principalization and resolution algorithms, blow-ups are permitted not only along smooth centers but also along Kummer centers, i.e., integrally closed ideals generated by regular and monomial parameters reflecting the snc structure. The snc divisor plays a dual role: it records the log structure and governs the admissibility of centers for subsequent blow-ups. This approach provides compatibility with log-smooth base change and refined functoriality.

5. Examples and Explicit Constructions

Resolution of a Plane Curve: For $X\subset \mathbb{P}^2$ given by the cuspidal cubic $y^2z = x^3$, with $E=\emptyset$, the effective log resolution algorithm with marked ideals yields two blow-ups at the origin in two charts, transforming the singular curve into a smooth strict transform intersecting the exceptional divisor transversely—thus producing an snc divisor structure (Bierstone et al., 10 Nov 2025, Abramovich et al., 17 Mar 2025).

Resolution for Theta Divisors: For the theta divisor $\Theta$ on the Jacobian of a hyperelliptic curve, the singular locus consists of Brill–Noether subvarieties $W^r_{g-1}$, and a sequence of blow-ups along these smooth centers—each of which corresponds locally to a determinantal variety—produces a strict transform $\tilde\Theta$ and exceptional divisors $Z_i$, all meeting transversely. The total transform

$$\pi_n^* \Theta = \tilde\Theta + \sum_{i=0}^{n-1} (n+1-i) Z_i$$

is a simple normal crossings divisor (Schnell et al., 2022).

6. Degree Bounds and Effectivity

Effective resolution results provide explicit degree and dimension bounds for the embedding of the final resolved pair $(X', E')$ in projective space. For input $(X, E) \subset \mathbb{P}^n$ of degrees $\leq d$, the final data can be taken as: $$n'(n,d) = 2^{O(n\log n)}, \quad d'(n,d) = (2d)^{2^{O(n)}}$$ with the resolution effected by at most $R(n, d) \leq M(n,d) + (M(n,d))!$ blow-ups, where $M(n,d) = d^{O(n^2)}$ (Bierstone et al., 10 Nov 2025). This establishes a primitive-recursive bound in the Grzegorczyk classification, demonstrating computability and effectivity of the algorithm, though the resulting complexity is infeasible for large $n$.

7. Applications and Extensions

The snc condition is crucial in:

• Construction of functorial or canonical log resolutions, especially over fields of characteristic zero (Temkin, 2023, Abramovich et al., 17 Mar 2025).
• Defining and analyzing log structures, log principalization, and relative semistable reduction in families.
• Applications to vanishing cycles, mixed Hodge modules, and multiplier ideals, where knowledge of intersection and singularity combinatorics is needed (Schnell et al., 2022).
• Mirror symmetry and log birational geometry, where snc divisors govern the behavior of degenerations and Gromov–Witten invariants (Corti et al., 14 Mar 2025).
• Algorithmic and computational approaches to resolving singularities in low-dimensional cases via computer algebra systems.

A plausible implication is that snc divisors provide the only environment in which resolution algorithms can be made fully functorial and compatible with logarithmic base change.

Summary Table: SNC Divisors and Their Roles

Context	Role of SNC Divisor	Key Fact/Definition
Classical Resolution	Final exceptional + strict transforms are snc	Local form: $V(t_1 \cdots t_d)$
Logarithmic Geometry	Model for divisorial log structure; log smoothness	$$M_X = \{ f \in \mathcal{O}_X : f|_{X\setminus D}\in \mathcal{O}_X^\times \}$$
Algorithmic Steps	Base condition for choice of blow-up centers	Centers must have nc with $E$

In summary, simple normal crossings divisors are indispensable in the structure theory and algorithms of singularities and their resolution, appearing as the uniquely tractable class of divisors under which effective, functorial, and explicit methods can be systematically developed and applied.