LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck pass

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Euler characteristic and normalization defect under a point blowup

Statement

Assume the Axiom of Choice. Let $k$ be a field, let $S$ be a regular surface proper over $k$ with an ample invertible sheaf, let $C \subseteq S$ be a reduced curve (an effective Cartier divisor), let $p$ be a closed point of $S$ with residue degree $r = [κ (p) : k]$ and let $m = {ord}_{m_{p}} (f) \geq 1$ be the order of a local equation $f$ of $C$ in $O_{S, p}$ , namely $f \in m_{p}^{m} ∖ m_{p}^{m + 1}$ . This is the intrinsic multiplicity, agreeing with Multiplicity of a hypersurface equation at a rational point in its affine rational-point setting. Let $π ⁣ : S^{'} \to S$ be the blowup of $p$ with exceptional curve $E$ and let $C^{'}$ be the strict transform of $C$ . Then $π^{*} C = C^{'} + m E$ as effective Cartier divisors on $S^{'}$ , and writing $χ_{k} (O_{X}) : = χ_{k} (X, O_{X})$ for the Euler characteristic of the structure sheaf (Euler characteristic of a coherent sheaf), one has $χ_{k} (O_{C^{'}}) = χ_{k} (O_{C}) + r (\binom{m}{2}) .$ Consequently, for the normalization defect $δ_{k}$ of Normalization defect delta of a reduced curve, $δ_{k} (C^{'}) = δ_{k} (C) - r (\binom{m}{2}) .$

Facts & Assumptions

Given: A field $k$ , a regular surface $S$ proper over $k$ with an ample invertible sheaf, a reduced effective Cartier divisor $C \subseteq S$ , a closed point $p$ with residue field $K = κ (p)$ of degree $r$ over $k$ , the multiplicity $m = {mult}_{p} (C) \geq 1$ , the blowup $π ⁣ : S^{'} \to S$ of $p$ with exceptional curve $E$ , and the strict transform $C^{'}$ . The Axiom of Choice is assumed as in the statement, inherited from the Proj and cohomology constructions (The Axiom of Choice).

[F1]

Total transform equals strict transform plus multiplicity times the exceptional divisor: For a reduced curve $C$ on a regular surface, a point blowup gives $π^{*} C = C^{'} + m E$ with $C^{'}$ the strict transform, and $C^{'}$ meets $E$ in the $0$ -cycle of degree $m$ over $K$ cut out by the degree- $m$ leading form. Both $C$ and $C^{'}$ are effective Cartier divisors (Cartier divisor).

[F2]

Point blowups of regular surfaces stay regular, with rational exceptional fibre over the residue field: $S^{'}$ is regular of pure dimension two, $E$ is an effective Cartier divisor isomorphic to $P_{K}^{1}$ , and $O_{E} (E) ≅ O_{P_{K}^{1}} (- 1)$ .

[F3]

Blowups of finite type ideals are locally H-projective, and proper and Strict transform of a closed subscheme: The blowup is proper over $S$ , hence proper over $k$ ; $C$ and $C^{'}$ are closed subschemes of $S$ and $S^{'}$ respectively, hence proper over $k$ ; $C^{'}$ is reduced because $C$ is reduced; and properness makes all these schemes of finite type over $k$ (Proper morphisms). Their affine coordinate rings are Noetherian because $k$ is Noetherian (Every algebra of finite type over a Noetherian ring is a Noetherian ring); thus their structure sheaves, and all finite locally free sheaves on them, are coherent (Coherent module sheaves).

[F4]

Pushforward and vanishing for point blowups on a surface: $π_{*} O_{S^{'}} = O_{S}$ and $R^{q} π_{*} O_{S^{'}} = 0$ for all $q > 0$ .

[F5]

Projection formula for invertible twists and Cohomology comparison when higher direct images vanish: For an invertible sheaf $L$ on $S$ the projection formula identifies $R^{q} π_{*} (π^{*} L) ≅ R^{q} π_{*} O_{S^{'}} \otimes L$ , so by [F4] the sheaf $π^{*} L$ has vanishing higher direct images and $π_{*} (π^{*} L) = L$ ; the vanishing-direct-image comparison gives $H^{q} (S^{'}, π^{*} L) ≅ H^{q} (S, L)$ for all $q$ , whence $χ_{k} (S^{'}, π^{*} L) = χ_{k} (S, L)$ since $S, S^{'}$ are proper over $k$ and the sheaves are coherent.

[F6]

Twisting the exact sequence of an effective Cartier divisor and Effective Cartier divisors give a short exact sequence: For an effective Cartier divisor $D$ on a scheme $X$ with closed immersion $i ⁣ : D \to X$ and an invertible sheaf $L$ there is a short exact sequence $0 ⟶ L (- D) ⟶ L ⟶ i_{*} (L ∣_{D}) ⟶ 0,$ where $L (- D) = L \otimes O_{X} (- D)$ is again invertible; for $L = O_{X}$ this is the standard sequence $0 \to O_{X} (- D) \to O_{X} \to i_{*} O_{D} \to 0$ .

[F7]

Euler characteristic is additive in short exact sequences: On a scheme proper over $k$ , the Euler characteristic of coherent sheaves is additive in short exact sequences.

[F8]

Closed immersion preserves cohomology and coherent pushforward: For a closed immersion $i ⁣ : Z \to X$ and a quasi-coherent $O_{Z}$ -module $F$ there are isomorphisms $H^{q} (Z, F) ≅ H^{q} (X, i_{*} F)$ for all $q \geq 0$ , and $i_{*} F$ is coherent when $X$ is locally Noetherian and $F$ is coherent.

[F9]

Euler characteristic of line bundles on a projective line over a finite field extension: For an invertible sheaf $M$ of degree $d$ over $K$ on $E ≅ P_{K}^{1}$ one has $χ_{k} (E, M) = r (1 + d)$ ; in particular a line bundle of degree $- j$ has $k$ -Euler characteristic $r (1 - j)$ .

[F10]

Normalization defect delta of a reduced curve, The normalization defect is an Euler characteristic and a weighted sum of local lengths, Normalization is unchanged under finite birational maps of reduced curves, A proper quasi-finite morphism is finite and Morphisms from a proper scheme to a separated one are proper: For a reduced proper curve $X$ over $k$ with normalization $\tilde{X}$ one has $δ_{k} (X) = χ_{k} (O_{\tilde{X}}) - χ_{k} (O_{X})$ ; a finite birational morphism of reduced curves induces an isomorphism of normalizations; and a morphism from a proper $k$ -scheme to a separated $k$ -scheme is proper, while a proper quasi-finite morphism is finite.

[F11]

The blowup is an isomorphism off the center: $π$ restricts to an isomorphism over $S ∖ {p}$ .

Proof

1.1F1F2F3

The curve $C$ is an effective Cartier divisor on the regular surface $S$ , so [F1] gives the divisor identity $π^{*} C = C^{'} + m E$ and shows that $C^{'}$ is again an effective Cartier divisor, meeting $E$ in a finite $0$ -cycle of degree $m$ over $K$ . By [F2] the exceptional curve is $E ≅ P_{K}^{1}$ with $O_{E} (E) ≅ O_{P_{K}^{1}} (- 1)$ , and by [F3] the schemes $S, S^{'}, C, C^{'}$ are proper and locally Noetherian of finite type over $k$ with coherent structure sheaves; $r = [K : k]$ is finite because $S$ is of finite type over $k$ .

2.1F6F7F8step 1.1

For $j = 0, 1, \dots, m$ put $L_{j} : = O_{S^{'}} (- π^{*} C + j E) : = O_{S^{'}} (- π^{*} C) \otimes O_{S^{'}} (E)^{\otimes j}$ , an invertible sheaf with $L_{m} = O_{S^{'}} (- C^{'})$ by the divisor identity of step 1.1. For $j \geq 1$ apply the twisted sequence of [F6] on $S^{'}$ to the effective Cartier divisor $E$ and the invertible sheaf $L_{j}$ : since $L_{j} (- E) = L_{j - 1}$ and writing $i ⁣ : E \to S^{'}$ for the closed immersion and $Q_{j} : = L_{j} ∣_{E}$ , one gets the short exact sequence $0 \to L_{j - 1} \to L_{j} \to i_{*} Q_{j} \to 0$ , whose three terms are coherent because $S^{'}$ is locally Noetherian. Additivity [F7] gives $χ_{k} (S^{'}, L_{j}) - χ_{k} (S^{'}, L_{j - 1}) = χ_{k} (S^{'}, i_{*} Q_{j})$ , and [F8] identifies the last term with $χ_{k} (E, Q_{j})$ .

2.2F6F7F8step 1.1

Apply the untwisted sequence of [F6] to the effective Cartier divisor $C$ on $S$ and to the effective Cartier divisor $C^{'}$ on $S^{'}$ , whose structure sheaves are coherent by step 1.1: $0 \to O_{S} (- C) \to O_{S} \to i_{*} O_{C} \to 0$ and $0 \to O_{S^{'}} (- C^{'}) \to O_{S^{'}} \to i_{*}^{'} O_{C^{'}} \to 0$ . Additivity [F7] and the identification $H^{q} (C, O_{C}) ≅ H^{q} (S, i_{*} O_{C})$ , respectively $H^{q} (C^{'}, O_{C^{'}}) ≅ H^{q} (S^{'}, i_{*}^{'} O_{C^{'}})$ , from [F8], give $χ_{k} (S, O_{S}) = χ_{k} (S, O_{S} (- C)) + χ_{k} (O_{C})$ and $χ_{k} (S^{'}, O_{S^{'}}) = χ_{k} (S^{'}, O_{S^{'}} (- C^{'})) + χ_{k} (O_{C^{'}})$ .

3.1F2F9algebrastep 2.1

The restriction $Q_{j} = L_{j} ∣_{E}$ is computed as follows. First, $O_{S^{'}} (j E) ∣_{E} = O_{E} (E)^{\otimes j}$ is isomorphic to $O_{P_{K}^{1}} (- j)$ by [F2]. Second, $(π^{*} O_{S} (- C)) ∣_{E} ≅ O_{E}$ : the morphism $π ∣_{E} ⁣ : E \to S$ factors as $E \to Spec K \to S$ , and the restriction of the invertible sheaf $O_{S} (- C)$ to the residue point is a free rank-one $K$ -module, whose pullback along $E \to Spec K$ is free of rank one. Hence $Q_{j} ≅ O_{P_{K}^{1}} (- j)$ , a line bundle of degree $- j$ over $K$ , and [F9] gives $χ_{k} (E, Q_{j}) = r (1 - j)$ .

4.1step 2.1step 3.1algebra

Summing the identities of step 2.1 over $j = 1, \dots, m$ and substituting step 3.1 gives $χ_{k} (S^{'}, L_{m}) - χ_{k} (S^{'}, L_{0}) = \sum_{j = 1}^{m} r (1 - j) = - r (\binom{m}{2})$ , that is, $χ_{k} (S^{'}, O_{S^{'}} (- C^{'})) = χ_{k} (S^{'}, π^{*} O_{S} (- C)) - r (\binom{m}{2})$ because $L_{0} = π^{*} O_{S} (- C)$ .

5.1F4F5step 4.1

By [F5] applied to the invertible sheaf $O_{S} (- C)$ the Euler characteristics agree: $χ_{k} (S^{'}, π^{*} O_{S} (- C)) = χ_{k} (S, O_{S} (- C))$ , and likewise $χ_{k} (S^{'}, O_{S^{'}}) = χ_{k} (S^{'}, π^{*} O_{S}) = χ_{k} (S, O_{S})$ . Combining with step 4.1, $χ_{k} (S^{'}, O_{S^{'}} (- C^{'})) = χ_{k} (S, O_{S} (- C)) - r (\binom{m}{2})$ .

6.1step 1.1step 2.2step 5.1

Subtracting the two identities of step 2.2 and substituting step 5.1 yields $χ_{k} (O_{C^{'}}) - χ_{k} (O_{C}) = [χ_{k} (S^{'}, O_{S^{'}}) - χ_{k} (S^{'}, O_{S^{'}} (- C^{'}))] - [χ_{k} (S, O_{S}) - χ_{k} (S, O_{S} (- C))] = χ_{k} (S^{'}, O_{S^{'}}) - χ_{k} (S, O_{S}) + r (\binom{m}{2}) = r (\binom{m}{2})$ , since $χ_{k} (S^{'}, O_{S^{'}}) = χ_{k} (S, O_{S})$ . This proves $χ_{k} (O_{C^{'}}) = χ_{k} (O_{C}) + r (\binom{m}{2})$ and, together with the divisor identity $π^{*} C = C^{'} + m E$ of step 1.1, the first assertions.

7.1F10F11step 1.1step 6.1

The morphism $C^{'} \to C$ induced by $π$ is proper: $C^{'}$ is a closed subscheme of the proper $k$ -scheme $S^{'}$ , hence proper over $k$ , and $C$ is separated over $k$ as a closed subscheme of the separated scheme $S$ ; by [F10] a morphism from a proper $k$ -scheme to a separated one is proper. It is quasi-finite: by [F11] it is an isomorphism over $C ∖ {p}$ , and over $p$ its fibre is the finite $0$ -cycle $C^{'} \cap E \subseteq E$ of step 1.1. It is birational: it is an isomorphism over the dense open $C ∖ {p}$ , $p$ being a closed point of the reduced curve $C$ . Hence $C^{'} \to C$ is finite by [F10], and [F10] identifies the normalizations of $C$ and $C^{'}$ over $C$ .

8.1F10step 6.1step 7.1∎

By step 1.1 both $C$ and $C^{'}$ are reduced proper curves over $k$ , so [F10] computes their defects on the common normalization $\tilde{C}$ : $δ_{k} (C) = χ_{k} (O_{\tilde{C}}) - χ_{k} (O_{C})$ and $δ_{k} (C^{'}) = χ_{k} (O_{\tilde{C}}) - χ_{k} (O_{C^{'}})$ . Subtracting and using step 6.1, $δ_{k} (C^{'}) - δ_{k} (C) = χ_{k} (O_{C}) - χ_{k} (O_{C^{'}}) = - r (\binom{m}{2})$ . Thus $π^{*} C = C^{'} + m E$ , $χ_{k} (O_{C^{'}}) = χ_{k} (O_{C}) + r (\binom{m}{2})$ and $δ_{k} (C^{'}) = δ_{k} (C) - r (\binom{m}{2})$ .

Remarks

The factor $r = [κ (p) : k]$ records the residue degree of the blown-up point: the successive quotients of the filtration are line bundles of degree $- j$ on a projective line over $κ (p)$ , and their $k$ -Euler characteristic is measured through $r$ .
Summing the identity over the singular points of a reduced curve on a regular surface gives the strictly decreasing invariant that drives the resolution algorithm; for $m = 1$ the correction vanishes, matching the fact that blowing up a regular point of a reduced curve does not change $χ_{k} (O_{C})$ .

Depends on

Used by

Dependency tree · two levels

170 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Ravi Vakil, Foundations of Algebraic Geometry, June 27, 2011 draft (author-hosted 'Early (out-of-date) version of The Rising Sea') (standard reference, not scraped)
J. S. Milne, Algebraic Geometry v6.10 (standard reference, not scraped)
The Stacks Project, Divisors, Sections 31.33-31.36 (Blowing up; Strict transform; Admissible blowups; Blowing up and flatness) (standard reference, not scraped)