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The blowup of the plane at the origin as an incidence scheme

Statement

Assume the Axiom of Choice, inherited from the Proj constructions used below (The Axiom of Choice). Let $B$ be a commutative ring with $1$ , $X = Spec B [x, y] = A_{B}^{2}$ and $I = (x, y) \subseteq B [x, y]$ . With homogeneous coordinates $u, v$ on $P_{B}^{1}$ , the blowup ${Bl}_{I} X$ of Blowup of a scheme along an ideal sheaf is $V (x v - y u) \subseteq X \times_{B} P_{B}^{1}$ , and the projection is its structural morphism. Its two standard charts are $Spec B [x, T]$ with $y = x T$ and $Spec B [y, U]$ with $x = y U$ ; their overlap inverts $T$ and $U$ with $T U = 1$ . The exceptional subscheme is $V (x)$ and $V (y)$ respectively and is isomorphic to $P_{B}^{1}$ . In particular this holds over any field $B = k$ .

Facts & Assumptions

Given: A commutative ring $B$ , the polynomial ring $B [x, y]$ , the ideal $I = (x, y)$ , the Rees algebra $R (I) = ⨁_{n \geq 0} I^{n} t^{n}$ (Rees algebra sheaf of a finite type ideal), the blowup ${Bl}_{I} Spec B [x, y] = {Proj}_{B [x, y]} R (I)$ (Blowup of a scheme along an ideal sheaf), and the projective line $P_{B}^{1}$ with standard charts $U_{0} = Spec B [t_{0}]$ , $U_{1} = Spec B [t_{1}]$ glued by $t_{0} \mapsto 1 / t_{1}$ (Relative projective space from standard charts).

[F1]

Affine blowup standard charts and overlaps: For $I = (f_{0}, \dots, f_{r})$ the standard opens $D_{+} (f_{i} t) = Spec A [I / f_{i}]$ cover ${Bl}_{I} Spec A$ , with overlap identifications given by the ratios $(f_{j} t) / (f_{i} t)$ .

[F2]

Affine blowup algebras: normal form, nonzerodivisors, reducedness, domains: For $a \in I$ the affine blowup algebra $A [I / a] = (R (I))_{(a)}$ has $I A [I / a] = a A [I / a]$ with $a$ a nonzerodivisor and $(A [I / a])_{a} = A_{a}$ , and is independent of the chosen generating set; for $I = (a, a_{1}, \dots, a_{r})$ the surjection $A [x_{1}, \dots, x_{r}] / (a x_{i} - a_{i}) \to A [I / a]$ , $x_{i} \mapsto a_{i} / a$ , has kernel the $a$ -power torsion.

[F3]

Relative projective space from standard charts, Projective space is Proj of a polynomial ring: $P_{B}^{1}$ is the gluing of its two standard charts, and for every commutative ring $A$ there is a canonical isomorphism $Proj A [u, v] ≅ P_{A}^{1}$ , natural in $A$ ; in particular $P_{B [x, y]}^{1} ≅ X \times_{B} P_{B}^{1}$ over $Spec B$ .

[F4]

Closed subschemes of projective space and saturated ideals: A homogeneous element $f$ of $A [u, v]$ of degree $d > 0$ cuts out the closed subscheme $V_{+} (f) = Proj (A [u, v] / (f)) ↪ P_{A}^{1}$ (Closed immersions of schemes), whose intersection with the standard open $D_{+} (u)$ is $Spec (A [u, v]_{(u)} / (f / u^{d}))$ (Standard opens of Proj).

[F5]

Exceptional subscheme of a blowup: The exceptional subscheme is the scheme-theoretic inverse image of the center, cut out by its inverse-image ideal.

Proof

1.1F1

The graded $B [x, y]$ -algebra homomorphism $B [x, y] [u, v] \to R (I)$ with $u \mapsto x t$ , $v \mapsto y t$ is surjective, since $I^{n} = (x, y)^{n}$ is generated by the monomials $x^{n - i} y^{i}$ , and its kernel is $(x v - y u)$ : the degree-zero kernel is zero, and for a homogeneous $F = \sum_{i = 0}^{n} c_{i} u^{n - i} v^{i}$ with $n \geq 1$ and $F (x, y) = 0$ , reduction modulo $x$ gives $c_{n} y^{n} \equiv 0 (m o d x)$ , so $c_{n} = x d$ because $y$ is a nonzerodivisor modulo $x$ in the polynomial ring $B [x, y]$ ; then $F - d v^{n - 1} (x v - y u) = u G$ with $G$ homogeneous of degree $n - 1$ , and $0 = F (x, y) = x G (x, y)$ forces $G (x, y) = 0$ because $x$ is a nonzerodivisor, so induction on $n$ gives $F \in (x v - y u)$ ; conversely $x v - y u \mapsto 0$ , and thus $R (I) ≅ B [x, y] [u, v] / (x v - y u)$ as graded $B [x, y]$ -algebras.

2.1F1F3F4step 1.1

Consequently ${Bl}_{I} X = {Proj}_{B [x, y]} (B [x, y] [u, v] / (x v - y u))$ , and by [F4] with $A = B [x, y]$ this is the closed subscheme $V (x v - y u)$ of $P_{B [x, y]}^{1} ≅ X \times_{B} P_{B}^{1}$ of [F3]; under this identification the structural morphism of the blowup is the projection to $X$ , and the standard charts of [F1] are $D_{+} (u) \cap V (x v - y u)$ and $D_{+} (v) \cap V (x v - y u)$ .

3.1F4step 2.1

Computing the two charts by [F4]: on $D_{+} (u)$ the dehomogenised equation is $x (v / u) - y = 0$ in $B [x, y, v / u]$ , so the chart ring is $B [x, T]$ with $T = v / u$ and $y = x T$ ; on $D_{+} (v)$ it is the quotient of $B [x, y, u / v]$ by $y (u / v) - x$ , that is $B [y, U]$ with $U = u / v$ and $x = y U$ . On the overlap $D_{+} (u v)$ both $u$ and $v$ are invertible, so $T = v / u$ and $U = u / v$ are mutually inverse units, $T U = 1$ , and the two chart rings agree on the overlap as localisations $B [x, T]_{T}$ and $B [y, U]_{U}$ under $T \mapsto U^{- 1}$ .

4.1F2F3F5step 3.1

By [F5], the exceptional subscheme is the inverse image of the origin $V (x, y)$ ; by [F2] its ideal on the first chart is $I B [x, T] = (x, x T) = (x)$ , so $E \cap D_{+} (u) = V (x) = Spec B [T]$ , and on the second chart it is $I B [y, U] = (y U, y) = (y)$ , so $E \cap D_{+} (v) = V (y) = Spec B [U]$ . On the overlap the rings $B [T]_{T} = B [T, T^{- 1}]$ and $B [U]_{U} = B [U, U^{- 1}]$ are identified by $T \mapsto U^{- 1}$ , which is exactly the gluing datum of the standard charts $U_{0}, U_{1}$ of $P_{B}^{1}$ in [F3]; hence $E ≅ P_{B}^{1}$ .

5.1F3step 2.1step 3.1step 4.1∎

Assembling steps 2.1, 3.1 and 4.1: the blowup is $V (x v - y u) \subseteq X \times_{B} P_{B}^{1}$ with the projection as structural morphism, its charts are $Spec B [x, T]$ with $y = x T$ and $Spec B [y, U]$ with $x = y U$ glued by $T U = 1$ , and the exceptional subscheme is $P_{B}^{1}$ ; no hypothesis on $B$ beyond commutativity was used, so the statement specialises to any field $B = k$ .

Remarks

The proof uses only that $x$ is a nonzerodivisor of $B [x, y]$ and that $y$ is a nonzerodivisor modulo $x$ ; neither $B$ a domain nor $B$ a field is needed, and the zero ring gives the empty blowup on both sides.
The equation $x v - y u = 0$ is the incidence relation of the point $(x, y)$ and the line $[u : v]$ , which is why the blowup is described as the incidence scheme; the strict transform computations of this page use this explicit presentation in the two charts.

Depends on

Used by

Nonflat base change of a blowup can fail Counterexample
First blowup of the cusp y²=x³ Example
First blowup of the node y²=x³+x² separates its branches Example
Resolving the rational map [x:y] at the origin Example
Total and strict transform of a line through the origin Example
Two charts of the blowup of the affine plane at the origin Example
Strict-transform equation by removing the maximal exceptional power Lemma
Blowing up a rational point of a smooth surface Theorem
Strict transforms of plane curves record tangent directions Theorem

Dependency tree · two levels

37 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)
The Stacks Project, Divisors, Sections 31.33-31.36 (Blowing up; Strict transform; Admissible blowups; Blowing up and flatness) (standard reference, not scraped)
Roman Bezrukavnikov et al., MIT 18.725 Algebraic Geometry (Fall 2015) consolidated lecture notes (standard reference, not scraped)