---
title: Weighted Blowup in Geometry and Algebra
url: https://www.emergentmind.com/topics/weighted-blowup
type: topic
---

# Weighted Blowup in Geometry and Algebra

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 [2504.11176] [2604.16698] [2501.08624].

## 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 \(\operatorname{Proj}(R_w)\) [2604.16698]. 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 \(\operatorname{Proj}_X(A^{\mathrm{ext}})\) after adjoining negative degrees [2501.08624]. In the Deligne–Mumford setting, weighted embeddings are described by decreasing ideals \(I_n\subset \mathcal O_X\) satisfying multiplicativity and finite generation conditions, and the weighted blowup is again \(\operatorname{Proj}_X(I_\bullet)\) [2310.15076].

A differential-geometric formulation replaces ideal filtrations by weightings on higher tangent bundles. If \(W\subset T^{(\infty)}M\) is a weighting along a closed submanifold \(N\subset M\), the weighted normal bundle \(\nu_W\) is obtained by quotienting \(W\) by indistinguishability with respect to functions of weighted order. The spherical weighted blowup is then
\[
\operatorname{Bl}_W(M):=(M\setminus N)\sqcup S\nu_W,\qquad 
S\nu_W:=(\nu_W\setminus N)/\mathbb R_{>0},
\]
while the projective weighted blowup replaces the \(\mathbb R_{>0}\)-quotient by the full \(\mathbb R\)-action and yields \((M\setminus N)\sqcup P\nu_W\) [2504.11176].

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 \(Z=\{x_1=\cdots=x_m=0\}\) has normal coordinates \(x_1,\dots,x_m\) with weights \(w_1,\dots,w_m\), the basic substitution is
\[
x_i=r^{w_i}u_i,\qquad r\ge 0,
\]
with \(u\) on a weighted sphere \(S_w\). In adapted charts one has anisotropic rescaling \( \lambda\cdot(v_1,\dots,v_m)=(\lambda^{w_1}v_1,\dots,\lambda^{w_m}v_m)\), and the Jacobian behavior matches these exponents [2504.11176]. In the orbifold Rees-algebra model, if \((\tilde x_1^{a_1},\dots,\tilde x_n^{a_n},\tilde t)\) are coordinates on the degeneration space, the blowdown map takes the form
\[
(\tilde x_1,\dots,\tilde x_n,\tilde t)\longmapsto
(x_1,\dots,x_n)=(\tilde t^{w_1}\tilde x_1,\dots,\tilde t^{w_n}\tilde x_n),
\]
and the exceptional divisor is cut out by \(t=0\) [2604.16698].

The exceptional divisor is the weighted analogue of the projectivized normal cone. In the orbifold/stack formulation it is
\[
E=b^{-1}(\operatorname{supp} Z_\bullet)\cong P(N(Z_\bullet))
=[(N(Z_\bullet)\setminus\{0\})/\mathbb G_m],
\]
a weighted projective bundle over the support of the center [2604.16698]. For Koszul-regular weighted centers on algebraic stacks, the weighted normal cone is
\[
C^w_{Z/X}=\operatorname{Spec}_X\!\left(\bigoplus_{n\ge 0} I_n/I_{n+1}\right),
\]
and the exceptional divisor is \(\operatorname{Proj}_X(\bigoplus I_n/I_{n+1})\), which under quasi-regularity becomes a twisted weighted projective stack bundle \(P_{Z_1}(N_{Z/X};d_0,\dots,d_n)\) [2501.08624]. In the Deligne–Mumford blow-down criterion, the converse direction is also available: if \(E\to Y\) is a weighted projective bundle and \(N_{E|X}\cong \mathcal O_E(-1)\otimes \pi^*L\), then \(X\) contracts to a smooth stack \(Z\), \(X\) is recovered as a weighted blowup of \(Z\) along \(Y\), and the contraction square is a pushout in algebraic stacks [2310.15076].

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 [2504.11176]. The elementary example with weights \((1,2)\) on \(\mathbb R^2\),
\[
x=r,\qquad y=r^2u
\quad\text{or}\quad
y=s^2,\qquad x=sv,
\]
already exhibits the weighted slope parameters that replace ordinary direction data [2504.11176].

## 3. Wonderful blowups, filtered manifolds, and functoriality

Weighted blowups extend from a single center to arrangements. For a building set \(\mathcal G\) of cleanly intersecting submanifolds equipped with compatible weightings \(W_G\), 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 [2504.11176]. Nests index the strata, and local coordinates include one control parameter \(t_N\ge 0\) 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 \(M\) carries a Lie filtration \(H_{-r}\supset\cdots\supset H_{-1}\supset H_0=0\), each diagonal \(\Delta_I\subset M^s\) acquires a canonical weighting \(W_{H_\bullet^s,\Delta_I}\). The resulting weighted Fulton–MacPherson blowup
\[
\operatorname{Conf}^{[s]}(M,H_\bullet)
\]
is a smooth manifold with corners. In local models indexed by forests, offsets \(\Delta x_{li}\) are scaled by products of control parameters raised to the filtration weights:
\[
\Delta x_{li}
=
(\Delta \hat x_{li})
\cdot
\prod_{N:\{l,\operatorname{par}(l)\}\subset N} t_N^{w_i}.
\]
For step-2 filtered manifolds such as contact manifolds, horizontal coordinates carry weight \(1\) and vertical coordinates weight \(2\), so two-point collision coordinates satisfy
\[
(x^{(i)}-x^{(j)})_h=r\,u_h,\qquad
(x^{(i)}-x^{(j)})_v=r^2\,u_v
\]
on the relevant weighted sphere [2504.11176].

A distinctive feature of the differential-geometric theory is the characterization of weightings by vanishing ideals. A closed connected embedded submanifold \(W\subset T^{(\infty)}M\) is a weighting iff its vanishing ideal is generated by lifted functions \(f^{(i)}\) whose lower lifts vanish on \(W\). One consequence is that clean intersections of weightings locally yield new weightings, which makes compatibility checks ideal-theoretic rather than coordinate-dependent [2504.11176]. Restricted functoriality is also available: a morphism of weighted building sets induces a smooth map on an open subspace of the corresponding blowups [2504.11176].

## 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 [2008.02169]. 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 [2512.01859].

For plane curves and singular hypersurfaces in regular two-dimensional schemes, the center is extracted from Hironaka’s characteristic polyhedron. If \(\operatorname{inv}_f(q)=(a_1,a_2)=(\nu,\nu\delta)\), the canonical monomial center is \(J=(x_1^{a_1},x_2^{a_2})\), with reduced center \(\bar J=(x_1^{1/w_1},x_2^{1/w_2})\) where \(\ell/w_i=a_i\). The stack-theoretic weighted blowup \([B_+/G_m]\to S\) 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 [2507.01232].

In Poisson geometry, weighted blowups are used to preserve compatibility with polyvector fields. For a regular weighted center \(Z_\bullet\) with weight sequence \(w\), a \(k\)-vector \(\xi\) lifts to the weighted blowup iff
\[
\operatorname{ord}_{Z_\bullet}(\xi)\ge -\gcd(w)
\quad\text{and}\quad
\operatorname{ord}_{N(Z_\bullet)}(\operatorname{lt}(\xi)\wedge E)\ge 0,
\]
where \(E\) is the weighted Euler field on the weighted normal bundle [2604.16698]. 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 [2604.16698].

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 \(cD/2\), discrepancy-\(2\) case realized as a weighted blowup inside a cyclic quotient of a smooth fivefold [1103.1182]. In a toric direction, Sankaran and Santos proved that for weighted blowups of \(\mathbf A^d\) with \(\varepsilon\)-log canonical singularities, the minimum weight is bounded by a constant depending only on \(\varepsilon\) and \(d\); in dimension \(4\), a terminal weighted blowup always has smallest weight at most \(32\), and at most \(6\) in all but finitely many cases [1911.06435].

## 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
\[
D_{qc}(\widetilde X)
=
\big\langle
j_*(p^*D_{qc}(Z_1)\otimes \mathcal O_E(1-\Sigma d_i)),
\dots,
j_*(p^*D_{qc}(Z_1)\otimes \mathcal O_E(-1)),
\pi^*D_{qc}(X)
\big\rangle,
\]
with \(\Sigma d_i-1\) exceptional components contributed by the exceptional divisor. When all weights are \(1\), this recovers Orlov’s blowup formula [2501.08624].

In symplectic orbifold geometry, weighted blowups are constructed by symplectic cutting. Blowing up a smooth point with weight vector \(m=(m_0,\dots,m_n)\) produces exceptional divisor \(Z\cong WP(m_0,\dots,m_n)\), and the normal orbibundle of \(Z\) on the blowup side is \(\mathcal O_m(-1)\) [1308.3634]. In this setting, primary orbifold Gromov–Witten invariants are unchanged under weighted blowup at a smooth point for genus \(g\le 1\) and \(n\ge 2\), and in all genera when the real dimension is \(4\) or \(6\) [1308.3634]. More generally, there is a weighted blowup correspondence between certain absolute orbifold Gromov–Witten invariants of \(X\) and certain relative invariants of the pair \((X_{\mathfrak a}\mid Z)\), expressed by an invertible lower-triangular transformation, and symplectic uniruledness is invariant under weighted blowup [1712.01478]. The full absolute orbifold Gromov–Witten theory of \(X_{\mathfrak a}\) can moreover be reconstructed from the theories of \(X\), the center \(S\), the exceptional divisor \(D_{\mathfrak a}\), the restriction map \(H^*_{CR}(X)\to H^*_{CR}(S)\), and \(c_1(\mathcal O_{D_{\mathfrak a}}(-1))\) [2009.06144].

Weighted blowups also admit explicit intersection-theoretic pushforward formulas. For a weighted blowup \(f:Y'\to Y\) of a smooth complete intersection center \(Z=\cap_i Z_i\) with weights \(w_i\), if \(E\) is the exceptional divisor, then
\[
f_* E^n
=
(-1)^{d+1}
\,h_{n-d}\!\left(\frac{Z_1}{w_1},\dots,\frac{Z_d}{w_d}\right)
\frac{Z_1}{w_1}\cdots \frac{Z_d}{w_d},
\]
and for any analytic function \(Q'(t)\) with coefficients pulled back from \(Y\),
\[
f_*Q'(E)
=
\sum_{n=1}^d
Q\!\left(\frac{Z_n}{w_n}\right)
\prod_{m\ne n}
\frac{\frac{Z_m}{w_m}}{\frac{Z_m}{w_m}-\frac{Z_n}{w_n}}.
\]
These formulas are used to compute generating functions of intersection numbers in weighted resolutions of F-theory models, including generic \(SU(5)\), \(F_4\), and \(Sp(6)\) Tate models [2305.00297].

## 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 \(\pi:\operatorname{Bl}_pM\to M\), while the adjective “weighted” refers not to the center but to the scalar curvature functional determined by positive weight functions \(v,w\) on the moment polytope:
\[
S_v(\omega)=v(\mu)S(\omega)-2\Delta(v(\mu))+\tfrac12\operatorname{Tr}(g\circ \operatorname{Hess}(v)(\mu)),
\qquad
S_{v,w}(\omega)=\frac{S_v(\omega)}{w(\mu)}.
\]
If \((M,\omega)\) is \((v,w)\)-weighted extremal and \(p\) is torus-fixed, relatively stable, and fixed by the weighted extremal field, then for sufficiently small \(\varepsilon>0\) the class
\[
[\omega_\varepsilon]=[\pi^*\omega]-\varepsilon^2[E]
\]
contains a \(T\)-invariant \((v,w)\)-weighted extremal metric [2304.08338]. 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 [2309.02279].

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 [2304.08338] [2309.02279]. 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.

Source: https://www.emergentmind.com/topics/weighted-blowup