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Blowing up a multiple point separates pairwise transverse components

Statement

Assume the Axiom of Choice. Let $S$ be a regular surface over a field $k$ and let $p$ be a closed point through which pass $s \geq 2$ distinct regular curves $Y_{1}, \dots, Y_{s}$ , pairwise meeting transversally at $p$ (contact orders one) and pairwise disjoint away from $p$ . Let $π ⁣ : S^{'} \to S$ be the blowup of $p$ with exceptional curve $E$ . Then the strict transforms $Y_{i}^{'}$ meet $E$ at $s$ distinct points, no three support curves meet at a point of $S^{'}$ (in particular at most two components pass through any point of $E$ ), and the only new intersections are the transverse intersections $Y_{i}^{'} \cap E$ at distinct points. If $s = 2$ the two strict transforms become disjoint.

Facts & Assumptions

Given: A regular surface $S$ over $k$ (a Noetherian scheme of dimension two regular at every point, in the sense of Contact order of two regular components at a point), a closed point $p \in S$ , distinct regular curves $Y_{1}, \dots, Y_{s}$ through $p$ with $s \geq 2$ , pairwise of contact order one at $p$ and pairwise disjoint away from $p$ , and the blowup $π ⁣ : S^{'} \to S$ of $p$ with exceptional curve $E$ .

[A1]

Choice. The Axiom of Choice is assumed, as in the statement; the cited suppliers used below are stated under it (The Axiom of Choice).

[F1]

Contact order of two regular components at a point: For distinct reduced curves $Y, Z$ through a closed point $p$ of a regular surface, the total contact order is the sum of the local lengths $n_{p} (Y, Z)$ and the definition records that $n_{p} (Y, Z) = 1$ if and only if $Y$ and $Z$ meet transversally at $p$ , meaning $p \in Y \cap Z$ , each of $Y$ and $Z$ is regular at $p$ , and their tangent lines are distinct one-dimensional subspaces of the two-dimensional $k (p)$ -vector space $m_{p} / m_{p}^{2}$ ; in that case the local contact order is computed in the local-equation form $n_{p} (Y, Z) = {length}_{O_{Y, p}} (O_{Y, p} / z O_{Y, p})$ for a local equation $z$ of $Z$ .

[F2]

A point blowup lowers pairwise contact order by one and separates transverse branches: Let $Y, Z$ be distinct regular curves through a closed point $p$ of a regular surface with contact order $n \geq 1$ , and let $Y^{'}, Z^{'}$ be their strict transforms under the blowup of $p$ with exceptional curve $E$ . If $n = 1$ , then $Y^{'}$ and $Z^{'}$ meet $E$ at distinct points and are disjoint near $E$ ; if $n > 1$ , the strict transforms meet at the point of $E$ corresponding to their common tangent direction with contact order $n - 1$ ; every intersection of a strict transform with $E$ has order one.

[F3]

Affine blowup standard charts and overlaps, Affine blowup algebras: normal form, nonzerodivisors, reducedness, domains, Flat base change for blowups, and failure without flatness, The exceptional divisor is the projectivized normal cone and associated graded ring of a regular local ring: Localizing the base at $p$ gives the charts $A [(x, y) / x]$ and $A [(x, y) / y]$ , with inverse ratio overlap. The exceptional curve is $Proj {gr}_{m} A$ , hence $P_{κ (p)}^{1}$ after choosing parameters, since $\dim A = 2$ at the contact point. The quotient presentations follow from the regular-sequence torsion calculation below.

[F4]

The blowup is an isomorphism off the center: The restriction of the blowup to the complement of the center is an isomorphism: $π ⁣ : π^{- 1} (S ∖ {p}) \to S ∖ {p}$ is an isomorphism of schemes.

[F6]

Strict transform of a closed subscheme: The strict transform of a closed subscheme is the scheme-theoretic closure of its inverse image minus the exceptional divisor; on a chart where the ideal of $E$ is invertible it is cut out by the saturation of the inverse-image ideal by the ideal of $E$ .

[F7]

dimension at most embedding dimension, regular local rings are domains and cohen macaulay, one dimensional regular local rings are dvrs and regular local quotient by parameter is regular: Regular local rings are domains, local dimension is at most embedding dimension, a regular parameter quotient is regular of dimension one less, and a regular hypersurface equation has multiplicity one.

[F8]

Total transform equals strict transform plus multiplicity times the exceptional divisor: For a reduced curve of multiplicity one at the blown-up point, the strict transform is given by dividing its equation by the exceptional equation; its intersection with $E$ is the divisor of its nonzero linear leading form.

Proof

1.1A1F1given

By [F1], since each pair $Y_{i}, Y_{j}$ with $i \neq j$ has contact order $n_{p} (Y_{i}, Y_{j}) = 1$ , every $Y_{i}$ is regular at $p$ and the tangent lines $T_{i} \subseteq m_{p} / m_{p}^{2}$ are pairwise distinct one-dimensional $k (p)$ -subspaces; moreover $Y_{i} \cap Y_{j} \subseteq {p}$ by the hypothesis that the curves are pairwise disjoint away from $p$ .

2.1F1F3F6F7F8step 1.1

Fix $i$ . Its regular prime quotient $A / P$ has cotangent dimension one, so choose $u \in P$ with nonzero cotangent class. The regular one-dimensional quotient $A / (u)$ is a DVR; its prime $P / (u)$ is zero because $(A / (u)) / (P / (u))$ still has dimension one. Thus $P = (u)$ and $u$ is a principal equation of the curve germ. Regularity makes its initial form a nonzero linear form $a X + b Y$ . Choose regular parameters $x, y$ so that $b \neq 0$ . The $x$ -chart is $B = A [T] / (x T - y)$ : reducing $x g = (x T - y) h$ modulo $x$ forces $h = x h_{1}$ , and cancellation proves the incidence quotient has no $x$ -power torsion. In this ring its strict transform is cut by $w = u / x$ , with $w m o d x = a + b T$ . Thus $Y_{i}^{'} \cap E$ is the single reduced point $T = - a / b$ . The other chart $A [U] / (y U - x)$ has equation $w^{'} = u / y$ with $w^{'} m o d y = a U + b$ ; it gives the same point if $a \neq 0$ , and none if $a = 0$ . This proves there are no other intersections with $E$ . At that point the ambient local ring has maximal ideal $(x, T + a / b)$ and prime chain $(0) ⊊ (x) ⊊ (x, T + a / b)$ ; it has dimension and embedding dimension two, so is regular with this cotangent basis, and $w$ has a nonzero coefficient on $T + a / b$ . Hence $Y_{i}^{'}$ is regular there and its tangent line differs from $E$ 's. Equivalently its quotient by $x$ is the residue field, giving contact length one.

3.1F2step 2.1

Applying [F2] with $n = 1$ to each pair $Y_{i}, Y_{j}$ , $i \neq j$ , the strict transforms meet $E$ at distinct points; with the uniqueness of step 2.1 this says $e_{i} \neq e_{j}$ whenever $i \neq j$ , so $Y_{1}^{'}, \dots, Y_{s}^{'}$ meet $E$ at the $s$ distinct points $e_{1}, \dots, e_{s}$ .

4.1F2F4step 1.1step 3.1

For $i \neq j$ the strict transforms $Y_{i}^{'}$ and $Y_{j}^{'}$ are disjoint: near $E$ this is [F2] with $n = 1$ , and outside $E$ the blowup restricts to an isomorphism of $S^{'} ∖ E$ with $S ∖ {p}$ by [F4], so a common point of $Y_{i}^{'}$ and $Y_{j}^{'}$ outside $E$ would map to a common point of $Y_{i}$ and $Y_{j}$ different from $p$ , which does not exist; hence $Y_{i}^{'} \cap Y_{j}^{'} = \emptyset$ , and in particular the two strict transforms are disjoint when $s = 2$ .

5.1F3F4step 2.1step 4.1

Consequently no three of the support curves $E, Y_{1}^{'}, \dots, Y_{s}^{'}$ meet at a point of $S^{'}$ : each $Y_{i}^{'}$ meets $E$ only in $e_{i}$ , the points $e_{i}$ are distinct, and the $Y_{i}^{'}$ are pairwise disjoint, so a point of $E$ lies on at most one strict transform and a point outside $E$ lies on at most one curve; every $e_{i}$ is a transverse intersection of $E$ with $Y_{i}^{'}$ by step 2.1. The only intersections not present before the blowup are these points $e_{i}$ : the original curves met one another only at $p$ , and each such intersection has been separated, while outside $E$ the blowup creates no new intersections because it is an isomorphism there [F4].

6.1step 3.1step 4.1step 5.1∎

Therefore $Y_{1}^{'}, \dots, Y_{s}^{'}$ meet $E$ at the $s$ distinct points $e_{1}, \dots, e_{s}$ , no three support curves meet at a point of $S^{'}$ , the only new intersections are the transverse intersections $Y_{i}^{'} \cap E = {e_{i}}$ at distinct points, and $Y_{1}^{'}$ and $Y_{2}^{'}$ are disjoint when $s = 2$ ; this proves every clause of the statement.

Depends on

Used by

Resolution of reduced plane curves by point blowups and the delta recurrence Theorem

Dependency tree · two levels

75 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)