Weighted Blow-Ups in Algebraic Geometry

• Weighted blow-ups are a birational geometric operation that replaces a smooth subvariety with a weighted projective bundle, capturing graded singularity data.
• They provide refined resolution methods by incorporating weight vectors, leading to precise intersection formulas and stacky exceptional divisors in moduli problems.
• Their construction via local charts and semi-orthogonal decompositions facilitates explicit computations in birational geometry, toroidal settings, and derived categories.

A weighted blow-up is a birational geometric operation that generalizes the classical blow-up by assigning a vector of positive integer weights to the local coordinate directions. This construction is crucial in birational geometry, resolution of singularities, and the theory of moduli spaces, especially in contexts where singularities or collision data possess nontrivial grading or higher-order structure. Weighted blow-ups appear ubiquitously in resolution algorithms, intersection theory, derived categories, minimal model program, and the study of moduli of algebraic stacks and stacks with group actions.

1. Definition and Local Construction

A weighted blow-up replaces a smooth subvariety (often a point or a submanifold) by a weighted projective bundle, encoding the specified directions and their assigned weights. Let $X$ be a smooth variety over a field of characteristic zero, and choose coordinates $x_1, \ldots, x_n$ near the center. Fix positive integers $w = (w_1,\ldots,w_n)$ with $\gcd(w_1,\ldots,w_n) = 1$. For a closed point $P$, define the sequence of weighted order ideals: $$I_m = \langle x_1^{\alpha_1}\cdots x_n^{\alpha_n} \mid \sum_{i=1}^{n} w_i \alpha_i \geq m \rangle.$$ The weighted blow-up is then constructed as the relative Proj: $$\mathrm{Bl}_w(X) = \mathrm{Proj}_X \bigoplus_{m\geq 0} I_m,$$ with exceptional divisor isomorphic to the weighted projective space $\mathbb{P}(w_1,\ldots,w_n)$ (Ishii, 2021, Sankaran et al., 2019, Andreatta, 2016).

Weighted blow-ups can be described in charts: on the $i$th chart $\{x_i \ne 0\}$, new coordinates $y_{ij}=x_j/x_i^{w_j/w_i}$ are used for $j\ne i$; the exceptional divisor corresponds to $x_i=0$. These charts are glued using transition functions respecting the weight data (Brais, 1 Dec 2025, Laface et al., 2017). For stacks or in the presence of group actions, the construction yields a stacky Proj (Li, 15 Jan 2025), and the exceptional divisor becomes a weighted projective bundle over the center (Arena et al., 2023).

2. Algebraic and Geometric Properties

Weighted blow-ups generalize classical blow-ups by accommodating centers with nontrivial grading and by yielding exceptional divisors that are weighted projective stacks rather than ordinary projective bundles. On a smooth ambient space, the blow-up morphism is proper, birational, and the resulting space may acquire quotient or stacky structure if the weights are not coprime (Andreatta, 2016, Arena et al., 2023).

Intersection theory on weighted blow-ups is governed by explicit formulas. For example, over a surface, the self-intersection of the exceptional divisor $E$ is $E^2 = -1/(ab)$ if the weights are $(a, b)$ (Laface et al., 2017). For a strict transform $C'$ of a curve $C$ passing through the center,

$(C')^2 = C^2 - \nu_{(a,b)}(C)^2/(ab),$

where $\nu_{(a,b)}(C)$ is the weighted multiplicity (Laface et al., 2017).

The Chow ring of a weighted blow-up is described in terms of the weighted Chern polynomial $P(t)$ of the normal bundle and explicit Gysin maps; the exceptional divisor is a weighted projective bundle with Chow ring $A^*(X)[t]/(P(t))$ (Arena et al., 2023). The stringy Chow ring, accounting for twisted sectors from stack structure, admits a decomposition indexed by the inertia of the automorphism group (Kuang et al., 1 Feb 2025).

3. Weighted Blow-Ups in Resolution Algorithms

Weighted blow-ups enable refined and often more efficient resolution procedures in birational geometry and singularity theory. Modern resolution algorithms (e.g., Abramovich–Temkin–Włodarczyk, Abramovich–Quek–Schober) use weighted blow-ups centered at subvarieties determined by invariants such as the Newton polyhedron or characteristic polyhedron of singularities (Brais, 1 Dec 2025, Abramovich et al., 1 Jul 2025).

Given a singular hypersurface $f(x) = 0$, the Newton polyhedron $NP(f)$ encodes the exponents of monomials. The minimal face $\Delta_w(f)$ with respect to a weight vector $w$ determines the center to be blown up; iteratively applying this weighted blow-up process yields a functorial resolution procedure with factorial reduction in complexity compared to history-dependent classical algorithms (Brais, 1 Dec 2025, Abramovich et al., 17 Mar 2025, Lee, 2020).

In the context of logarithmic and toroidal geometry, weighted toroidal blow-ups generalize the construction to centers involving monoidal or Kummer ideals, enabling functorial logarithmic resolutions as well (Quek, 2020).

4. Applications to Moduli Spaces and Birational Geometry

Weighted blow-ups are fundamental in the study of moduli of stacks and spaces with singularities. For example, the moduli stack $\overline{\mathcal{M}}_{1,n}$ of stable $n$-pointed genus-one curves is realized as a weighted blow-up (with weights 4, 6) of the stack of pseudo-stable curves along a stacky center, with exceptional divisor the boundary of elliptic tails (Arena et al., 2023).

In the birational geometry of higher-dimensional varieties, weighted blow-ups serve as canonical local models for divisorial contractions with terminal or canonical singularities. The "lifting" result shows that if a divisorial contraction restricts to a weighted blow-up on a divisor, then the contraction itself must be realized as a weighted blow-up (Andreatta, 2016).

The boundedness of minimal weights for ε-log canonical and terminal singularities is established: for instance, in dimension four, the minimal weight is at most 32 for terminal blow-ups, with the vast majority bounded by 6 (Sankaran et al., 2019). This result addresses conjectures of Birkar and establishes explicit bounds through the classification of empty simplices.

5. Derived Categories and Semi-Orthogonal Decompositions

Weighted blow-ups induce semi-orthogonal decompositions of derived categories, extending Orlov's classic results for ordinary blow-ups to the weighted and stacky context (Li, 15 Jan 2025). For a weighted blow-up of a smooth algebraic stack $X$ along a Koszul-regular center, there is a decomposition: $$D^\mathrm{b}(\mathrm{Bl}^w X) = \langle j_*(p^* D^\mathrm{b}(Z_1) \otimes \mathcal{O}_E(r)),\ldots, \pi^* D^\mathrm{b}(X) \rangle,$$ where the range and the structure are determined by the weights. This structure is analogous to Beilinson-type decompositions and closely tracks the stacky and grading data.

Weighted blow-ups thus preserve and enhance the functorial and categorical structure necessary for applications in modern algebraic geometry, particularly in derived and categorical geometry.

6. Configuration Spaces, Filtered Manifolds, and Weighted Arrangements

Weighted blow-ups naturally arise in the construction of compactified configuration spaces with collision data of higher order, such as in filtered or sub-Riemannian manifolds and jet spaces. The weighted structure is required to correctly capture the compatibility of infinitesimal collision data with the underlying filtration (Gootjes-Dreesbach, 15 Apr 2025). The general framework covers both spherical and projective weighted blow-ups, proves smoothness under clean intersection assumptions, and establishes local models for configuration spaces incorporating higher-order structures.

Applications also extend to arrangements of submanifolds with assigned weightings: clean local intersection and compatible filtrations ensure that the weighted blow-up process preserves smoothness and the desired geometric structure (Gootjes-Dreesbach, 15 Apr 2025).

7. Illustrative Examples and Explicit Constructions

Typical examples include the (weighted) resolution of plane curve singularities, such as $f(y,x) = y^2 - x^5= 0$, where the weighted blow-up along the center determined by the characteristic polyhedron achieves strict reduction of order in a single step (Abramovich et al., 1 Jul 2025, Brais, 1 Dec 2025, Abramovich et al., 17 Mar 2025). The method applies equally to higher-dimensional situations, including the explicit resolution of singularities of threefolds with bounds on the number of weighted blow-ups required to compute minimal log discrepancies (Ishii, 2021).

In toric and stack-theoretic frameworks, weighted blow-ups are realized as stacky Proj constructions and have stacky exceptional divisors, introducing twisted sectors in Chow and stringy cohomology rings (Arena et al., 2023, Kuang et al., 1 Feb 2025).

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Weighted blow-ups have become a central tool in modern algebraic geometry, both for their role in streamlined resolution algorithms and for the rich geometric and categorical structures they induce, enabling explicit and computable models for birational contractions, moduli problems, and the structure of singularities.