04.07.02 · algebraic-geometry / projective

Blowup

shipped3 tiersLean: partial

Anchor (Master): Hironaka 1964 *Resolution of Singularities*; Kollár *Lectures on Resolution of Singularities*; Hauser *The Hironaka Theorem*

Intuition [Beginner]

A blowup surgically replaces a point on a variety with the set of all directions through that point. Imagine standing at a point $P$ on a surface and refusing to identify which direction you're looking — you replace $P$ with the entire space of directions emanating from $P$ . The result is a new variety with a " $P^{1}$ of directions" sitting where $P$ used to be.

Why bother? Because singular points of varieties — places where the variety is not smooth — often resolve into smooth pieces once you blow them up. The cone with a sharp point at the origin becomes a smooth surface after replacing the cone point with the circle of generating lines. Heisuke Hironaka proved in 1964 that every singular variety in characteristic 0 can be resolved by repeated blowups along smooth centres. He won the Fields Medal for this, and the proof runs 217 pages.

Blowups also let you separate curves that pass through the same point: two distinct lines through the origin become two non-intersecting curves on the blowup, distinguished by their direction.

Visual [Beginner]

The plane $A^{2}$ with the origin replaced by a copy of $P^{1}$ — the circle of directions through the origin.

Worked example [Beginner]

The blowup of the affine plane $A^{2} = {(x, y)}$ at the origin is the subset of $A^{2} \times P^{1}$ defined by the equation $x \cdot v = y \cdot u$ , where $[u : v]$ are homogeneous coordinates on $P^{1}$ . Above any point $(a, b) \neq = (0, 0)$ , the equation forces $[u : v] = [a : b]$ — exactly one point. Above $(0, 0)$ , the equation is automatic, and the entire $P^{1}$ sits there.

Concretely: the line $y = 2 x$ passes through the origin with slope 2. On the blowup it becomes the curve $(x, 2 x, [1 : 2])$ — its image at the origin is the single direction $[1 : 2]$ . The line $y = 3 x$ becomes $(x, 3 x, [1 : 3])$ , hitting a different direction. Two lines that met at the origin downstairs are now disjoint upstairs: each line has its own direction and they no longer share a point.

This is the essence of blowing up — separating data that was previously collapsed into a single point.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $X$ be a Noetherian scheme and $I \subseteq O_{X}$ a coherent ideal sheaf, defining a closed subscheme $Z \subseteq X$ . The blowup of $X$ along $Z$ is the scheme

Bl_{Z} X := Proj_{X} n \geq 0 ⨁ I^{n},

the relative Proj of the Rees algebra $⨁_{n \geq 0} I^{n} \subseteq O_{X} [t]$ of the ideal sheaf, equipped with a structural morphism $π : Bl_{Z} X \to X$ .

Universal property. $π$ is the universal morphism among $X$ -schemes $f : Y \to X$ such that the inverse image ideal sheaf $f^{- 1} I \cdot O_{Y}$ is an invertible $O_{Y}$ -module: for every such $f$ , there exists a unique $X$ -morphism $Y \to Bl_{Z} X$ factoring $f$ .

Exceptional divisor. The preimage $E := π^{- 1} (Z) = Proj_{Z} ⨁_{n \geq 0} I^{n} / I^{n + 1}$ is an effective Cartier divisor on $Bl_{Z} X$ , the exceptional divisor. Off the exceptional divisor, $π$ is an isomorphism: $π^{- 1} (X ∖ Z) \sim X ∖ Z$ .

Local picture. When $X = Spec (R)$ is affine and $I = (f_{1}, \dots, f_{r})$ , the blowup is the closure in $X \times P^{r - 1}$ of the graph

{(x, [f_{1} (x) : \dots : f_{r} (x)]) : x \in X ∖ Z},

cut out by the equations $f_{i} \cdot u_{j} = f_{j} \cdot u_{i}$ for $1 \leq i, j \leq r$ , where $[u_{1} : \dots : u_{r}]$ are homogeneous coordinates on $P^{r - 1}$ .

Examples.

Blowup of $A^{n}$ at the origin. With $I = (x_{1}, \dots, x_{n})$ , the blowup $A^{n} \subset A^{n} \times P^{n - 1}$ is defined by $x_{i} u_{j} = x_{j} u_{i}$ . The exceptional divisor is ${0} \times P^{n - 1}$ , a copy of projective space of dimension $n - 1$ .
Blowup of a smooth subvariety. If $Z \subset X$ is smooth of codimension $c$ in a smooth variety, then $Bl_{Z} X$ is again smooth, and the exceptional divisor $E \to Z$ is a projective bundle: $E ≅ P (N_{Z / X})$ , the projectivisation of the normal bundle.
Blowup of a node on a plane curve. The nodal cubic $y^{2} = x^{2} (x + 1)$ has a node at the origin. Blowing up the origin separates the two branches into smooth disjoint curves on the blowup.

Strict transform. Given a closed subscheme $Y \subseteq X$ with $Y \neq \subseteq Z$ , the strict transform $Y \subseteq Bl_{Z} X$ is the closure of $π^{- 1} (Y ∖ Z)$ in $Bl_{Z} X$ . The total transform $π^{- 1} (Y)$ in general decomposes as $Y \cup m E$ where $m$ is the multiplicity of $Y$ along $Z$ .

Key theorem with proof [Intermediate+]

Theorem (universal property of the blowup). Let $X$ be a Noetherian scheme, $Z \subseteq X$ a closed subscheme with ideal sheaf $I$ , and $π : Bl_{Z} X \to X$ the blowup. Then $π^{- 1} I \cdot O_{Bl_{Z} X}$ is an invertible sheaf, and for any morphism $f : Y \to X$ such that $f^{- 1} I \cdot O_{Y}$ is invertible, there exists a unique $X$ -morphism $f : Y \to Bl_{Z} X$ with $π \circ f = f$ .

Proof. Write $R := ⨁_{n \geq 0} I^{n}$ for the Rees algebra. By construction $Bl_{Z} X = Proj_{X} (R)$ .

Step 1 — invertibility on the blowup. On $Proj_{X} (R)$ , the sheaf $O (1)$ is invertible by the standard Proj construction. The image of the natural map $π^{*} I \to π^{*} R_{1} \to O_{Bl_{Z} X} (1)$ is a surjection from an invertible sheaf onto an invertible sheaf, hence an isomorphism: $π^{- 1} I \cdot O_{Bl_{Z} X} ≅ O_{Bl_{Z} X} (1)$ , an invertible sheaf.

Step 2 — universal factorisation. Suppose $f : Y \to X$ has $f^{- 1} I \cdot O_{Y} =: L$ invertible. The inclusion $I ↪ O_{X}$ pulls back to a surjection $f^{*} I ↠ L$ . Since $L$ is invertible, this surjection corresponds to an $X$ -morphism $Y \to P_{X} (I)$ — specifically, into the closure of the graph of "directions of $I$ ." That closure is exactly $Bl_{Z} X = Proj_{X} (Sym^{∙} I) / \sim= Proj_{X} (R)$ (the normalisation of the symmetric algebra by the Rees relations).

Concretely, on an affine chart $X = Spec (R)$ with $I = (f_{1}, \dots, f_{r})$ , an invertible $L$ on $Y$ together with a surjection $O_{Y}^{r} ↠ L$ via $e_{i} \mapsto f_{i}$ determines a unique morphism $Y \to P^{r - 1}$ . The relations $f_{i} u_{j} = f_{j} u_{i}$ on $Bl_{Z} X$ are automatically satisfied by this morphism (the $f_{i}$ are scalar multiples of a single generator of $L$ locally), so the morphism factors through the blowup.

Step 3 — uniqueness. The factoring morphism is determined by the surjection $f^{*} I \to L$ , which is itself determined by $f$ and the invertibility data — no additional choice. Hence $f$ is unique. $□$

The universal property characterises the blowup up to canonical isomorphism. It is the minimal surgery on $X$ that turns $I$ into an invertible ideal sheaf.

Bridge. The construction here builds toward later units of the strand, where the same pattern is taken up at higher structure. The defining pattern appears again in those units in a sharpened form, where the local data is glued or quotiented. Putting these together, the foundational insight is that the data of this unit gives the structural signature that the rest of the strand reads off.

Exercises [Intermediate+]

Exercise 3 (medium, proof).

Show that the blowup of $A^{2}$ at the origin separates the two branches of the nodal curve $y^{2} = x^{2} (x + 1)$ at the node.

Hint

Substitute $y = xv$ in the chart $u \neq = 0$ and factor.

Answer

In the chart $u \neq = 0$ (with $y = xv$ ), the nodal equation $y^{2} = x^{2} (x + 1)$ becomes $(xv)^{2} = x^{2} (x + 1)$ , i.e., $x^{2} v^{2} = x^{2} (x + 1)$ . Factoring: $x^{2} (v^{2} - x - 1) = 0$ .

The factor $x^{2} = 0$ is the exceptional divisor (with multiplicity 2). The factor $v^{2} - x - 1 = 0$ is the strict transform: a smooth conic in the $(x, v)$ plane meeting the exceptional divisor $x = 0$ at $v^{2} = 1$ , i.e., at $v = \pm 1$ .

The two branches of the original node had tangent directions $y = \pm x$ at the origin, corresponding to slopes $v = \pm 1$ . The blowup picks out exactly these two points $(0, \pm 1)$ on the exceptional divisor — separating the branches into two smooth points on the strict transform. The strict transform is smooth, while the original was nodal. $□$

Exercise 4 (medium, conceptual).

Why is the exceptional divisor of a blowup of a smooth variety along a smooth centre always a projective bundle?

Hint

Locally near $Z$ , the ideal sheaf is generated by codimension-many regular parameters whose normal directions form a vector bundle.

Answer

Let $Z \subset X$ be a smooth closed subvariety of codimension $c$ in a smooth variety $X$ . Locally near $Z$ , by smoothness, the ideal sheaf $I$ is generated by $c$ regular parameters $f_{1}, \dots, f_{c}$ — equivalently, $I / I^{2}$ is a locally free $O_{Z}$ -module of rank $c$ , the conormal sheaf. Its dual is the normal sheaf $N_{Z / X}$ , also locally free of rank $c$ .

The exceptional divisor of the blowup is

E = Proj_{Z} n \geq 0 ⨁ I^{n} / I^{n + 1} = Proj_{Z} Sym^{∙} (I / I^{2}) .

Since $I / I^{2}$ is locally free, $Sym^{∙}$ is the symmetric algebra of a vector bundle, and $Proj_{Z} Sym^{∙}$ of a vector bundle equals its projectivisation. Concretely:

E = P (N_{Z / X}^{\lor}) \to Z,

a projective bundle over $Z$ with fibres $P^{c - 1}$ . So the exceptional divisor records the directions normal to $Z$ — not just the points of $Z$ but the projective bundle of normal lines.

Exercise 5 (medium, proof).

Show that the blowup $π : Bl_{Z} X \to X$ is a projective birational morphism.

Hint

Projectivity comes from $Proj$ ; birationality from the isomorphism off $Z$ .

Answer

Projectivity. The blowup is locally $Proj$ of a finitely generated graded $O_{X}$ -algebra, hence projective over $X$ by the standard Proj construction. The relative Proj of a finitely generated graded algebra is a projective $X$ -scheme.

Birationality. Off the closed subscheme $Z$ , the ideal sheaf $I$ equals $O_{X}$ , so $I ∣_{X ∖ Z} = O_{X ∖ Z}$ . The Rees algebra restricts to $O_{X ∖ Z} [t]$ , and $Proj$ of a polynomial ring in one variable is just the base scheme. So $π^{- 1} (X ∖ Z) ≅ X ∖ Z$ — the morphism is an isomorphism off the centre.

Hence $π$ is projective and a birational morphism (an isomorphism on a dense open). $□$

Exercise 6 (hard, proof).

Compute the Picard group of the blowup of a smooth surface $X$ at a single point $P$ , in terms of $Pic (X)$ .

Hint

Pic of the blowup is generated by pullback classes and the exceptional divisor class.

Answer

Let $X = Bl_{P} X$ and $E$ the exceptional divisor (a copy of $P^{1}$ ). The pullback $π^{*} : Pic (X) \to Pic (X)$ is injective.

Claim: $Pic (X) ≅ Pic (X) \oplus Z \cdot [E]$ .

Sketch. Surjectivity: every Cartier divisor on $X$ decomposes as a pullback (its proper transform in $X ∖ P$ extends back to $X$ as a pullback class, modulo a multiple of $E$ ). The exceptional divisor $E$ is not a pullback (its image in $X$ is the single point $P$ , which is not a divisor on the surface).

Independence: $π^{*} (D) = n E$ in $Pic (X)$ would mean $D \cdot O_{X} = π_{*} (O_{X} (n E))$ . But $π_{*} O_{X} (n E) = O_{X}$ for all $n \geq 0$ (since $E$ projects to a point), and the dual gives torsion-free for $n < 0$ . Either way, $D = 0$ in $Pic (X)$ , forcing also $n = 0$ from the pullback restriction.

Intersection theory. $E^{2} = - 1$ (the self-intersection of an exceptional curve on a smooth blowup), $E \cdot π^{*} (D) = 0$ for all $D \in Pic (X)$ , and $π^{*} (D_{1}) \cdot π^{*} (D_{2}) = D_{1} \cdot D_{2}$ . So the intersection form on $Pic (X)$ is the orthogonal sum of $Pic (X)$ and the rank-1 lattice $⟨ E ⟩$ with $E^{2} = - 1$ . $□$

Exercise 7 (hard, conceptual).

State Hironaka's theorem on resolution of singularities, and explain why blowups along smooth centres suffice in characteristic 0.

Hint

The theorem produces a smooth variety birational to any given variety via repeated blowups.

Answer

Hironaka's theorem (1964). Let $X$ be an algebraic variety over a field $k$ of characteristic zero. There exists a finite sequence of blowups along smooth centres

X_{n} \to X_{n - 1} \to \dots \to X_{1} \to X_{0} = X,

such that $X_{n}$ is smooth and the composite morphism $X_{n} \to X$ is a proper birational map that is an isomorphism over the smooth locus of $X$ .

Why smooth centres suffice in characteristic 0. Hironaka's induction is on a complex invariant — the Hironaka invariant — measuring the "worst" singularity at any point in terms of multiplicity, residue invariants, and exceptional history. He shows that one can always find a smooth subvariety contained in the singular locus along which a blowup strictly decreases this invariant. Because the invariant takes values in a well-ordered set, the process terminates after finitely many steps.

The hypothesis of characteristic zero is essential: in characteristic $p$ , the same induction fails because of wild phenomena — purely inseparable extensions and Frobenius-twisted singularities. Resolution of singularities in positive characteristic remains open in dimensions $\geq 4$ (proved by Hironaka in $\leq 3$ , by de Jong via alterations — non-birational — for all dimensions, with full birational resolution still elusive).

Hironaka's 1964 paper runs 217 pages; he was awarded the Fields Medal in 1970 for the work. The theorem is the foundational result enabling the minimal model program and modern birational geometry. $□$

Lean formalization [Intermediate+]

lean_status: partial — Mathlib has the underlying $Proj$ infrastructure and graded ring theory; the named blowup construction and its universal property are not yet formalised in full.

[object Promise]

Advanced results [Master]

Hironaka's theorem on resolution of singularities (1964). Every algebraic variety over a field of characteristic 0 admits a resolution of singularities — a proper birational morphism from a smooth variety, obtained by a finite sequence of blowups along smooth centres. The proof, occupying 217 pages of Annals of Mathematics 79, introduced the Hironaka invariant — a sophisticated lexicographic well-ordering of singularity types — and showed that a single blowup along an appropriately chosen smooth centre strictly decreases it. The work was simplified by Bierstone-Milman, Encinas-Villamayor, Włodarczyk, and Kollár, but the essential strategy remains Hironaka's.

Equisingular and embedded resolution. Hironaka actually proved a stronger embedded version: given a closed subscheme $Y \subset X$ with $X$ smooth, there exists a sequence of blowups along smooth centres in the singular locus of $Y$ resolving $Y$ and making it transverse to (or contained in) a normal-crossings divisor. This stronger form is what's used in birational geometry.

Weak factorisation theorem (Włodarczyk-Abramovich-Karu-Matsuki, 2002). Any birational map between smooth projective varieties in characteristic 0 can be factored as a sequence of blowups and blowdowns along smooth centres. So blowups generate the entire birational equivalence relation among smooth projective varieties.

Castelnuovo's contraction theorem. On a smooth projective surface, an irreducible curve $E$ with $E ≅ P^{1}$ and $E^{2} = - 1$ can be contracted to a smooth point: there exists a morphism $X \to X^{'}$ with $X^{'}$ smooth, contracting $E$ and being an isomorphism elsewhere. Conversely every blowup of a smooth point on a smooth surface produces such a $(- 1)$ -curve. Castelnuovo's theorem is the converse to surface blowup and is foundational for the classification of surfaces.

Failure in positive characteristic. Resolution of singularities is open in characteristic $p > 0$ for varieties of dimension $\geq 4$ . Hironaka proved it in dimensions $\leq 3$ . de Jong (1996) proved a weaker statement — resolution by alterations (generically finite morphisms instead of birational) — in all characteristics. The failure in characteristic $p$ stems from wild ramification, inseparability, and the Frobenius automorphism producing pathological singularity types not amenable to Hironaka's induction.

Toric resolutions. For toric varieties, resolution of singularities reduces to combinatorics of fans: every singular cone admits a regular subdivision producing a smooth toric blowup. This combinatorial resolution is constructive and works in all characteristics.

Synthesis. This construction generalises the pattern fixed in 04.02.01 (scheme), with the symmetric data replaced by its skew or twisted analogue. Read in the opposite direction, the construction is dual to the metric story: complements and orthogonality are taken with respect to the bilinear datum of this unit, not a metric, and the resulting category of subobjects is the one the rest of the strand classifies. The central insight is that this datum identifies algebra with geometry: functions become vector fields, subspaces become quotients, and invariants become cohomology classes — and that identification is the engine driving every theorem downstream.

Full proof set [Master]

The universal property is proved in the formal-definition section. Hironaka's theorem itself is one of the deepest theorems in algebraic geometry; full proofs run book-length (Kollár's Lectures on Resolution of Singularities, Princeton 2007; Hauser's exposition in the Bulletin of the AMS, 2003). A self-contained proof is beyond the scope of this unit and is stated without proof — see Hironaka Annals of Math 79 (1964); Kollár Lectures; Hauser The Hironaka Theorem on Resolution of Singularities, Bulletin AMS 40 (2003), 323–403.

The proof of $E^{2} = - 1$ for the exceptional curve of a single-point blowup of a smooth surface uses the projection formula $π_{*} (E \cdot π^{*} D) = (π_{*} E) \cdot D = 0 \cdot D = 0$ , combined with the local computation in coordinates. The Picard-group decomposition $Pic (X) = Pic (X) \oplus Z \cdot [E]$ follows by the localisation sequence in $K$ -theory or by direct cocycle analysis.

Connections [Master]

Scheme 04.02.01 — blowup is a construction internal to the category of schemes, producing a new scheme from a closed subscheme.
Projective space 04.07.01 — the blowup is locally a closed subscheme of a product with projective space, reflecting the "directions" interpretation.
Cartier divisor 04.05.04 — the exceptional divisor is the canonical effective Cartier divisor on the blowup; the universal property says blowups make pulled-back ideal sheaves Cartier.
Coherent sheaf 04.06.02 — the Rees algebra is a sheaf of graded $O_{X}$ -algebras built from coherent ideal sheaves.
Picard group 04.05.02 — blowing up a smooth point on a smooth surface adds a $Z$ -summand to the Picard group, generated by the exceptional curve class.
Riemann-Roch theorem for curves 04.04.01 — strict transforms of curves under blowup change their genus and degree predictably; key in classification of surfaces.
Moduli of curves 04.10.01 — blowups of $\overline{M_{g}}$ produce alternative compactifications relevant to enumerative geometry.

Historical & philosophical context [Master]

Oscar Zariski began the modern study of resolution of singularities in the 1930s, proving the case of surfaces (1939) and threefolds (1944) using algebraic methods involving normalisation and Bertini-type arguments. He pursued resolution in higher dimensions for the rest of his career without success: the obstruction lay in finding the right invariant to induct on.

Heisuke Hironaka, Zariski's student, solved the problem in characteristic 0 in his 1964 paper Resolution of singularities of an algebraic variety over a field of characteristic zero, published in Annals of Mathematics 79, 109–326. The argument runs 217 pages and introduces the Hironaka invariant — a refined lexicographic measurement of singularity complexity — together with an inductive scheme showing that blowups along carefully chosen smooth centres strictly decrease this invariant. Hironaka was awarded the Fields Medal in 1970. The work is widely regarded as one of the deepest theorems in algebraic geometry; subsequent decades produced multiple simplifications (Bierstone-Milman, Encinas-Villamayor, Włodarczyk, Kollár), but the essential Hironaka strategy remains.

The blowup construction itself is older — present in classical Italian algebraic geometry (Cremona, Castelnuovo, Enriques) for surfaces, where it figured prominently in the minimal models programme. Castelnuovo's contraction theorem (1899) — a smooth surface contains a $(- 1)$ -curve $E$ with $E ≅ P^{1}$ and $E^{2} = - 1$ if and only if it is the blowup of another smooth surface at a point — gave the classical converse. Zariski reformulated blowups schematically in the 1940s; Grothendieck systematised the general construction via the relative Proj of the Rees algebra in EGA II (1961).

The universal property — blowup is the minimal surgery making an ideal sheaf invertible — is Grothendieck's reformulation, now standard. Cartier divisors, line bundles, and the Picard scheme make the universal property a theorem about representability of a moduli functor.

In the minimal model programme (Mori, Reid, Kollár, Mori-Mukai, Birkar-Cascini-Hacon-McKernan, 2010), blowups and their inverses (contractions) are the elementary surgeries on higher-dimensional varieties. The theorem of Birkar-Cascini-Hacon-McKernan (2010, J. Amer. Math. Soc.) establishing existence of minimal models for varieties of general type uses iterated blowups and contractions in essential ways.

Resolution of singularities in characteristic $p > 0$ remains open in dimensions $\geq 4$ . de Jong (1996, Publ. Math. IHES 83) proved resolution by alterations — generically finite morphisms instead of birational — in all characteristics. Whether full birational resolution holds in positive characteristic is one of the major open problems in algebraic geometry.

Bibliography [Master]

Hironaka, H., Resolution of singularities of an algebraic variety over a field of characteristic zero, Annals of Mathematics 79 (1964), 109–326.
Hironaka, H., On the theory of birational blowing-up, Harvard PhD thesis (1960).
Zariski, O., The reduction of the singularities of an algebraic surface, Annals of Mathematics 40 (1939), 639–689.
Hartshorne, R., Algebraic Geometry, Springer 1977 — §II.7.
Vakil, R., The Rising Sea: Foundations of Algebraic Geometry — §22.
Kollár, J., Lectures on Resolution of Singularities, Princeton University Press 2007.
Hauser, H., The Hironaka Theorem on Resolution of Singularities, Bulletin AMS 40 (2003), 323–403.
Włodarczyk, J., Simple Hironaka resolution in characteristic zero, J. Amer. Math. Soc. 18 (2005), 779–822.
de Jong, A. J., Smoothness, semi-stability and alterations, Publ. Math. IHES 83 (1996), 51–93.
Bierstone-Milman, Canonical desingularization in characteristic zero, Invent. Math. 128 (1997), 207–302.
Eisenbud-Harris, The Geometry of Schemes, Springer 2000 — §IV.2.

Prerequisites

04.02.01
04.07.01
04.05.04

Tier anchors

beginner: Strogatz-style: replacing a point with the directions through it
intermediate: Hartshorne §II.7; Vakil §22; Eisenbud-Harris *Geometry of Schemes* §IV.2
master: Hironaka 1964 *Resolution of Singularities*; Kollár *Lectures on Resolution of Singularities*; Hauser *The Hironaka Theorem*

References

TODO_REF
Hartshorne — Algebraic Geometry · §II.7, blowing up
TODO_REF
Vakil — The Rising Sea · §22, blowups
TODO_REF
Hironaka — Resolution of singularities of an algebraic variety over a field of characteristic zero · Annals of Mathematics 79 (1964)
TODO_REF
Kollár — Lectures on Resolution of Singularities · Princeton 2007

Lean module

Codex.AlgebraicGeometry.Projective.Blowup

Reviewer

TBD

Estimated time

beginner: 15m
intermediate: 40m
master: 70m