Weighted Blowup in Geometry and Algebra

• Weighted blowup is a modification technique that replaces isotropic scaling with anisotropic scaling by using weighted filtrations and Rees algebras.
• It constructs exceptional divisors as weighted projectivized normal cones or bundles, ensuring precise control over singularity structures in various geometric settings.
• This approach underpins resolution of singularities, supports functorial constructions in stacks, and enables effective intersection-theoretic computations.

A weighted blowup is a blowup-type modification in which the normal directions to a center are assigned non-uniform weights, so that isotropic scaling is replaced by anisotropic scaling and the exceptional divisor records weighted approach directions rather than ordinary radial directions. In algebraic, stack-theoretic, symplectic, and differential-geometric settings, the construction is governed by a weighted filtration or weight vector and produces an exceptional object that is typically a weighted projectivized normal cone, a weighted projective bundle, or a weighted sphere; this contrasts with the ordinary blowup, whose exceptional divisor is the usual projectivized or spherical normal bundle (Gootjes-Dreesbach, 15 Apr 2025, Lapointe et al., 17 Apr 2026, Li, 15 Jan 2025).

1. Defining constructions

In the algebraic and stack-theoretic literature, a weighted center is encoded by a filtration of ideals or, equivalently, a weighted Rees algebra. On an orbifold $X$, a weighted center may be given étale locally by weighted coordinates $(x_1^{a_1},\dots,x_n^{a_n})$, or by a nonincreasing weight sequence $w=(w_1,\dots,w_n)$ with $w_i=1/a_i\in \mathbb{Q}_{\ge 0}$. The induced filtration is

$$I_\lambda=\{f\in \mathcal O_X\mid \operatorname{ord}_w(f)\ge \lambda\},\qquad \operatorname{ord}_w(x^J)=\sum_i w_i j_i,$$

and the weighted blowup is obtained from the extended Rees algebra by a $\mathbb G_m$-quotient. In the integer-weight case one may also write a graded weighted Rees algebra $R_w=\bigoplus_{m\ge 0} I_m T^m$ and realize the blowup as $\operatorname{Proj}(R_w)$ (Lapointe et al., 17 Apr 2026). For algebraic stacks, a weighted closed immersion is similarly encoded by a graded quasi-coherent Rees algebra $$A=\bigoplus_{n\ge 0} I_n t^n\subset \mathcal O_X[t]$$, and the weighted blowup is the stacky Proj

$\operatorname{Bl}_A X:=\operatorname{Proj}_X(A),$

or equivalently $(x_1^{a_1},\dots,x_n^{a_n})$0 after adjoining negative degrees (Li, 15 Jan 2025). In the Deligne–Mumford setting, weighted embeddings are described by decreasing ideals $(x_1^{a_1},\dots,x_n^{a_n})$1 satisfying multiplicativity and finite generation conditions, and the weighted blowup is again $(x_1^{a_1},\dots,x_n^{a_n})$2 (Arena et al., 2023).

A differential-geometric formulation replaces ideal filtrations by weightings on higher tangent bundles. If $(x_1^{a_1},\dots,x_n^{a_n})$3 is a weighting along a closed submanifold $(x_1^{a_1},\dots,x_n^{a_n})$4, the weighted normal bundle $(x_1^{a_1},\dots,x_n^{a_n})$5 is obtained by quotienting $(x_1^{a_1},\dots,x_n^{a_n})$6 by indistinguishability with respect to functions of weighted order. The spherical weighted blowup is then

$(x_1^{a_1},\dots,x_n^{a_n})$7

while the projective weighted blowup replaces the $(x_1^{a_1},\dots,x_n^{a_n})$8-quotient by the full $(x_1^{a_1},\dots,x_n^{a_n})$9-action and yields $w=(w_1,\dots,w_n)$0 (Gootjes-Dreesbach, 15 Apr 2025).

Across these formulations, the common structure is a filtration by weighted order, an associated graded object, and a quotient construction that substitutes weighted normal directions for the original center.

2. Local models, exceptional divisors, and singularities

Locally, weighted blowups are governed by weighted substitutions. In smooth differential geometry, if $w=(w_1,\dots,w_n)$1 has normal coordinates $w=(w_1,\dots,w_n)$2 with weights $w=(w_1,\dots,w_n)$3, the basic substitution is

$w=(w_1,\dots,w_n)$4

with $w=(w_1,\dots,w_n)$5 on a weighted sphere $w=(w_1,\dots,w_n)$6. In adapted charts one has anisotropic rescaling $w=(w_1,\dots,w_n)$7, and the Jacobian behavior matches these exponents (Gootjes-Dreesbach, 15 Apr 2025). In the orbifold Rees-algebra model, if $w=(w_1,\dots,w_n)$8 are coordinates on the degeneration space, the blowdown map takes the form

$w=(w_1,\dots,w_n)$9

and the exceptional divisor is cut out by $w_i=1/a_i\in \mathbb{Q}_{\ge 0}$0 (Lapointe et al., 17 Apr 2026).

The exceptional divisor is the weighted analogue of the projectivized normal cone. In the orbifold/stack formulation it is

$w_i=1/a_i\in \mathbb{Q}_{\ge 0}$1

a weighted projective bundle over the support of the center (Lapointe et al., 17 Apr 2026). For Koszul-regular weighted centers on algebraic stacks, the weighted normal cone is

$w_i=1/a_i\in \mathbb{Q}_{\ge 0}$2

and the exceptional divisor is $w_i=1/a_i\in \mathbb{Q}_{\ge 0}$3, which under quasi-regularity becomes a twisted weighted projective stack bundle $w_i=1/a_i\in \mathbb{Q}_{\ge 0}$4 (Li, 15 Jan 2025). In the Deligne–Mumford blow-down criterion, the converse direction is also available: if $w_i=1/a_i\in \mathbb{Q}_{\ge 0}$5 is a weighted projective bundle and $w_i=1/a_i\in \mathbb{Q}_{\ge 0}$6, then $w_i=1/a_i\in \mathbb{Q}_{\ge 0}$7 contracts to a smooth stack $w_i=1/a_i\in \mathbb{Q}_{\ge 0}$8, $w_i=1/a_i\in \mathbb{Q}_{\ge 0}$9 is recovered as a weighted blowup of $$I_\lambda=\{f\in \mathcal O_X\mid \operatorname{ord}_w(f)\ge \lambda\},\qquad \operatorname{ord}_w(x^J)=\sum_i w_i j_i,$$0 along $$I_\lambda=\{f\in \mathcal O_X\mid \operatorname{ord}_w(f)\ge \lambda\},\qquad \operatorname{ord}_w(x^J)=\sum_i w_i j_i,$$1, and the contraction square is a pushout in algebraic stacks (Arena et al., 2023).

Weighted singularities are often intrinsic rather than accidental. In the projective differential-geometric model, orbifold charts show that parity of the nonzero weights controls smoothness: if all nonzero weights are odd, the projective weighted blowup is a smooth manifold; if all are even, it is a smooth manifold with boundary; mixed parity yields genuine orbifold singularities (Gootjes-Dreesbach, 15 Apr 2025). The elementary example with weights $$I_\lambda=\{f\in \mathcal O_X\mid \operatorname{ord}_w(f)\ge \lambda\},\qquad \operatorname{ord}_w(x^J)=\sum_i w_i j_i,$$2 on $$I_\lambda=\{f\in \mathcal O_X\mid \operatorname{ord}_w(f)\ge \lambda\},\qquad \operatorname{ord}_w(x^J)=\sum_i w_i j_i,$$3,

$$I_\lambda=\{f\in \mathcal O_X\mid \operatorname{ord}_w(f)\ge \lambda\},\qquad \operatorname{ord}_w(x^J)=\sum_i w_i j_i,$$4

already exhibits the weighted slope parameters that replace ordinary direction data (Gootjes-Dreesbach, 15 Apr 2025).

3. Wonderful blowups, filtered manifolds, and functoriality

Weighted blowups extend from a single center to arrangements. For a building set $$I_\lambda=\{f\in \mathcal O_X\mid \operatorname{ord}_w(f)\ge \lambda\},\qquad \operatorname{ord}_w(x^J)=\sum_i w_i j_i,$$5 of cleanly intersecting submanifolds equipped with compatible weightings $$I_\lambda=\{f\in \mathcal O_X\mid \operatorname{ord}_w(f)\ge \lambda\},\qquad \operatorname{ord}_w(x^J)=\sum_i w_i j_i,$$6, the weighted graph blowup is defined as the closure of the diagonal embedding into the product of the individual weighted blowups. Under separation of factors and uniform alignment of the weightings along every nest, the resulting weighted blowup is a smooth manifold with corners, and the blow-down map is smooth and proper (Gootjes-Dreesbach, 15 Apr 2025). Nests index the strata, and local coordinates include one control parameter $$I_\lambda=\{f\in \mathcal O_X\mid \operatorname{ord}_w(f)\ge \lambda\},\qquad \operatorname{ord}_w(x^J)=\sum_i w_i j_i,$$7 for each nest element, so that corner faces encode successive weighted collisions.

This framework was introduced in part to construct configuration spaces of filtered manifolds. If $$I_\lambda=\{f\in \mathcal O_X\mid \operatorname{ord}_w(f)\ge \lambda\},\qquad \operatorname{ord}_w(x^J)=\sum_i w_i j_i,$$8 carries a Lie filtration $$I_\lambda=\{f\in \mathcal O_X\mid \operatorname{ord}_w(f)\ge \lambda\},\qquad \operatorname{ord}_w(x^J)=\sum_i w_i j_i,$$9, each diagonal $\mathbb G_m$0 acquires a canonical weighting $\mathbb G_m$1. The resulting weighted Fulton–MacPherson blowup

$\mathbb G_m$2

is a smooth manifold with corners. In local models indexed by forests, offsets $\mathbb G_m$3 are scaled by products of control parameters raised to the filtration weights: $\mathbb G_m$4 For step-2 filtered manifolds such as contact manifolds, horizontal coordinates carry weight $\mathbb G_m$5 and vertical coordinates weight $\mathbb G_m$6, so two-point collision coordinates satisfy

$\mathbb G_m$7

on the relevant weighted sphere (Gootjes-Dreesbach, 15 Apr 2025).

A distinctive feature of the differential-geometric theory is the characterization of weightings by vanishing ideals. A closed connected embedded submanifold $\mathbb G_m$8 is a weighting iff its vanishing ideal is generated by lifted functions $\mathbb G_m$9 whose lower lifts vanish on $R_w=\bigoplus_{m\ge 0} I_m T^m$0. One consequence is that clean intersections of weightings locally yield new weightings, which makes compatibility checks ideal-theoretic rather than coordinate-dependent (Gootjes-Dreesbach, 15 Apr 2025). Restricted functoriality is also available: a morphism of weighted building sets induces a smooth map on an open subspace of the corresponding blowups (Gootjes-Dreesbach, 15 Apr 2025).

4. Resolution of singularities and birational extraction

Weighted blowups have become a standard tool in characteristic-zero resolution algorithms. In the Abramovich–Temkin–Włodarczyk framework, implemented algorithmically by Lee, weighted resolution proceeds by repeatedly blowing up along centers determined by a canonical invariant and independent of the history of previous blowups; this “history-free” feature sharply distinguishes it from older algorithms (Lee, 2020). A later graphical approach based on Newton graphs and systems of parameters extends the plane-curve constructions to arbitrary codimension and is described as yielding a factorial reduction in complexity compared with the original ATW procedure (Brais, 1 Dec 2025).

For plane curves and singular hypersurfaces in regular two-dimensional schemes, the center is extracted from Hironaka’s characteristic polyhedron. If $R_w=\bigoplus_{m\ge 0} I_m T^m$1, the canonical monomial center is $R_w=\bigoplus_{m\ge 0} I_m T^m$2, with reduced center $R_w=\bigoplus_{m\ge 0} I_m T^m$3 where $R_w=\bigoplus_{m\ge 0} I_m T^m$4. The stack-theoretic weighted blowup $R_w=\bigoplus_{m\ge 0} I_m T^m$5 then has the property that the order of the proper transform strictly decreases at every point above the center; iteration gives an embedded resolution by tame Artin stacks (Abramovich et al., 1 Jul 2025).

In Poisson geometry, weighted blowups are used to preserve compatibility with polyvector fields. For a regular weighted center $R_w=\bigoplus_{m\ge 0} I_m T^m$6 with weight sequence $R_w=\bigoplus_{m\ge 0} I_m T^m$7, a $R_w=\bigoplus_{m\ge 0} I_m T^m$8-vector $R_w=\bigoplus_{m\ge 0} I_m T^m$9 lifts to the weighted blowup iff

$\operatorname{Proj}(R_w)$0

where $\operatorname{Proj}(R_w)$1 is the weighted Euler field on the weighted normal bundle (Lapointe et al., 17 Apr 2026). This criterion underlies functorial orbifold reductions of singularities for Poisson subvarieties in smooth Poisson threefolds, stopping only at two explicitly identified normal forms: Du Val surface points and non-nilpotent curve singularities (Lapointe et al., 17 Apr 2026).

Weighted blowups also appear in classification results. Kawakita proved that every threefold divisorial contraction to a non-Gorenstein point is a weighted blow-up, with the remaining $\operatorname{Proj}(R_w)$2, discrepancy-$\operatorname{Proj}(R_w)$3 case realized as a weighted blowup inside a cyclic quotient of a smooth fivefold (Kawakita, 2011). In a toric direction, Sankaran and Santos proved that for weighted blowups of $\operatorname{Proj}(R_w)$4 with $\operatorname{Proj}(R_w)$5-log canonical singularities, the minimum weight is bounded by a constant depending only on $\operatorname{Proj}(R_w)$6 and $\operatorname{Proj}(R_w)$7; in dimension $\operatorname{Proj}(R_w)$8, a terminal weighted blowup always has smallest weight at most $\operatorname{Proj}(R_w)$9, and at most $$A=\bigoplus_{n\ge 0} I_n t^n\subset \mathcal O_X[t]$$0 in all but finitely many cases (Sankaran et al., 2019).

5. Categorical, enumerative, and intersection-theoretic consequences

Weighted blowups carry substantial categorical structure. For a Koszul-regular weighted center on an algebraic stack, the quasi-coherent derived category of the weighted blowup admits a semi-orthogonal decomposition

$$A=\bigoplus_{n\ge 0} I_n t^n\subset \mathcal O_X[t]$$1

with $$A=\bigoplus_{n\ge 0} I_n t^n\subset \mathcal O_X[t]$$2 exceptional components contributed by the exceptional divisor. When all weights are $$A=\bigoplus_{n\ge 0} I_n t^n\subset \mathcal O_X[t]$$3, this recovers Orlov’s blowup formula (Li, 15 Jan 2025).

In symplectic orbifold geometry, weighted blowups are constructed by symplectic cutting. Blowing up a smooth point with weight vector $$A=\bigoplus_{n\ge 0} I_n t^n\subset \mathcal O_X[t]$$4 produces exceptional divisor $$A=\bigoplus_{n\ge 0} I_n t^n\subset \mathcal O_X[t]$$5, and the normal orbibundle of $$A=\bigoplus_{n\ge 0} I_n t^n\subset \mathcal O_X[t]$$6 on the blowup side is $$A=\bigoplus_{n\ge 0} I_n t^n\subset \mathcal O_X[t]$$7 (He et al., 2013). In this setting, primary orbifold Gromov–Witten invariants are unchanged under weighted blowup at a smooth point for genus $$A=\bigoplus_{n\ge 0} I_n t^n\subset \mathcal O_X[t]$$8 and $$A=\bigoplus_{n\ge 0} I_n t^n\subset \mathcal O_X[t]$$9, and in all genera when the real dimension is $\operatorname{Bl}_A X:=\operatorname{Proj}_X(A),$0 or $\operatorname{Bl}_A X:=\operatorname{Proj}_X(A),$1 (He et al., 2013). More generally, there is a weighted blowup correspondence between certain absolute orbifold Gromov–Witten invariants of $\operatorname{Bl}_A X:=\operatorname{Proj}_X(A),$2 and certain relative invariants of the pair $\operatorname{Bl}_A X:=\operatorname{Proj}_X(A),$3, expressed by an invertible lower-triangular transformation, and symplectic uniruledness is invariant under weighted blowup (Chen et al., 2017). The full absolute orbifold Gromov–Witten theory of $\operatorname{Bl}_A X:=\operatorname{Proj}_X(A),$4 can moreover be reconstructed from the theories of $\operatorname{Bl}_A X:=\operatorname{Proj}_X(A),$5, the center $\operatorname{Bl}_A X:=\operatorname{Proj}_X(A),$6, the exceptional divisor $\operatorname{Bl}_A X:=\operatorname{Proj}_X(A),$7, the restriction map $\operatorname{Bl}_A X:=\operatorname{Proj}_X(A),$8, and $\operatorname{Bl}_A X:=\operatorname{Proj}_X(A),$9 (Chen et al., 2020).

Weighted blowups also admit explicit intersection-theoretic pushforward formulas. For a weighted blowup $(x_1^{a_1},\dots,x_n^{a_n})$00 of a smooth complete intersection center $(x_1^{a_1},\dots,x_n^{a_n})$01 with weights $(x_1^{a_1},\dots,x_n^{a_n})$02, if $(x_1^{a_1},\dots,x_n^{a_n})$03 is the exceptional divisor, then

$(x_1^{a_1},\dots,x_n^{a_n})$04

and for any analytic function $(x_1^{a_1},\dots,x_n^{a_n})$05 with coefficients pulled back from $(x_1^{a_1},\dots,x_n^{a_n})$06,

$(x_1^{a_1},\dots,x_n^{a_n})$07

These formulas are used to compute generating functions of intersection numbers in weighted resolutions of F-theory models, including generic $(x_1^{a_1},\dots,x_n^{a_n})$08, $(x_1^{a_1},\dots,x_n^{a_n})$09, and $(x_1^{a_1},\dots,x_n^{a_n})$10 Tate models (Arena et al., 2023).

6. Terminological variants in weighted extremal geometry

A distinct usage of “weighted blowup” appears in Kähler geometry. In Hallam’s work on weighted extremal metrics, the birational modification is the ordinary blowup $(x_1^{a_1},\dots,x_n^{a_n})$11, while the adjective “weighted” refers not to the center but to the scalar curvature functional determined by positive weight functions $(x_1^{a_1},\dots,x_n^{a_n})$12 on the moment polytope: $(x_1^{a_1},\dots,x_n^{a_n})$13 If $(x_1^{a_1},\dots,x_n^{a_n})$14 is $(x_1^{a_1},\dots,x_n^{a_n})$15-weighted extremal and $(x_1^{a_1},\dots,x_n^{a_n})$16 is torus-fixed, relatively stable, and fixed by the weighted extremal field, then for sufficiently small $(x_1^{a_1},\dots,x_n^{a_n})$17 the class

$(x_1^{a_1},\dots,x_n^{a_n})$18

contains a $(x_1^{a_1},\dots,x_n^{a_n})$19-invariant $(x_1^{a_1},\dots,x_n^{a_n})$20-weighted extremal metric (Hallam, 2023). In the sequel on weighted K-stability, this blowup theorem is used to prove that a weighted extremal manifold is relatively weighted K-polystable, and that a weighted cscK manifold is weighted K-polystable, allowing singular degenerations (Hallam, 2023).

This terminological divergence is explicit: in that literature, “weighted blowup” does not mean an algebraic weighted blowup, but an ordinary blowup studied inside a weighted extremal or weighted scalar curvature framework (Hallam, 2023, Hallam, 2023). A plausible implication is that “weighted blowup” now functions as a family resemblance term across several areas: anisotropic birational modification, stacky weighted projectivization, weighted collision resolution, and blowup constructions coupled to weighted analytic functionals.