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Nonvanishing charts of sections of an ample linearization are affine

Statement

Assume AC inherited from the embedding suppliers. Let $X$ be a complex projective variety, let $L$ be an ample $G$ -linearized invertible sheaf, and let $σ \in Γ (X, L^{\otimes n})$ be a global section with $n \geq 1$ . Then the nonvanishing locus $X_{σ} = {x \in X : σ (x) \neq 0}$ is an affine open subset of $X$ ; if $σ$ is $G$ -invariant, then $X_{σ}$ is $G$ -stable. Consequently the charts $X_{σ}$ , for $σ \in Γ (X, L^{\otimes n})^{G}$ and $n \geq 1$ , form an open cover of $X^{s s} (L)$ by affine $G$ -stable subsets. Here, for any complex affine algebraic group $G$ , $X^{s s} (L)$ denotes the union of these invariant nonvanishing loci; for reductive $G$ this agrees with Semistable and stable points for a linearization.

Facts & Assumptions

Given: A complex projective variety $X$ with an algebraic action of the complex affine algebraic group $G$ , an ample $G$ -linearized invertible sheaf $L$ on $X$ , a global section $σ \in Γ (X, L^{\otimes n})$ , and a $G$ -equivariant closed immersion $i : X ↪ P (V)$ with $i^{*} O (1) ≅ L^{\otimes m}$ as $G$ -linearized invertible sheaves for some $m \geq 1$ .

[F1]

Equivariant embedding. There are $m \geq 1$ , a finite-dimensional rational $G$ -module $V$ and a $G$ -equivariant closed immersion $i : X ↪ P (V)$ with $i^{*} O (1) ≅ L^{\otimes m}$ as $G$ -linearized invertible sheaves, obtained from the complete linear system $∣ L^{\otimes m} ∣$ ; in particular $Γ (X, L^{\otimes m k}) ≅ Γ (X, (L^{\otimes m})^{\otimes k})$ for every $k \geq 0$ . (An ample linearization embeds equivariantly after a positive power, Linearizations of tensor powers and the equivariant section ring)

[F2]

Projective charts and high-degree lifting. A closed subscheme of projective space has affine standard charts $Y \cap D_{+} (F)$ . The ideal sheaf is coherent by Coherent sheaves on a locally Noetherian scheme and Closed immersion preserves cohomology and coherent pushforward. The sequence $0 \to I_{Y} (k) \to O_{P (V)} (k) \to i_{*} O_{Y} (k) \to 0$ and Long exact sequence of sheaf cohomology make restriction surjective when $H^{1} (I_{Y} (k)) = 0$ , which holds for large $k$ by Serre vanishing. The source space consists of homogeneous degree- $k$ forms by Cohomology of O(d) on projective space (also for $P^{0}$ and $k \geq 0$ ). No surjectivity in every degree is assumed. (Serre vanishing for coherent sheaves and ample twists, Closed subschemes of projective space and saturated ideals, Standard opens are affine)

[F3]

Nonvanishing loci. For invertible sheaves the nonvanishing locus of a section is open, and intersecting with an affine open gives an affine open; for sections $σ, τ$ one has $X_{σ \otimes τ} = X_{σ} \cap X_{τ}$ . (A line-bundle section cuts an affine open inside an affine scheme, Absolute ampleness by affine section opens)

Proof

technique · direct

1.1F3algebra

Fix the equivariant closed immersion of [F1], so that $L^{\otimes m} ≅ i^{*} O (1)$ . For a local trivialization of $L$ in which $σ$ corresponds to a regular function $f$ , the section $σ^{\otimes m}$ corresponds to $f^{m}$ ; hence $σ^{\otimes m}$ and $σ$ have the same nonvanishing locus, $X_{σ^{\otimes m}} = X_{σ}$ (this is the local computation of the nonvanishing locus, valid for arbitrary sections, not only invariant ones).

1.2F1algebra

If $σ$ is $G$ -invariant, then for every $g \in G$ the equality $g \cdot σ = σ$ means $σ (g x) = g σ (x)$ in the fibre of $L^{\otimes n}$ at $g x$ ; since the fibre map is a $C$ -linear isomorphism, $σ (x) = 0$ if and only if $σ (g x) = 0$ . Hence $X_{σ}$ is $G$ -stable.

2.1F1F2F3step 1.1

Under the embedding of step 1.1, $σ^{\otimes m}$ is a section of $O_{X} (n)$ . For the coherent ideal sheaf $I_{X}$ in $P (V)$ , Serre vanishing gives $H^{1} (P (V), I_{X} (n q)) = 0$ for sufficiently large $q$ . The ideal-sheaf exact sequence then makes restriction of degree- $n q$ forms onto $H^{0} (X, O_{X} (n q))$ surjective. Lift $σ^{\otimes m q}$ to such a form $F$ . Its nonvanishing locus equals $X_{σ}$ , since taking a positive power does not change vanishing in a line fibre. Hence $X_{σ} = X \cap D_{+} (F)$ , a closed subscheme of the standard affine Proj chart, and is affine. No projective-normality assumption is used.

3.1step 2.1step 1.2∎

Every $x \in X^{s s} (L)$ has, by the invariant-section union specified in the Statement, an invariant section $σ \in Γ (X, L^{\otimes n})^{G}$ with $σ (x) \neq 0$ , hence lies in the affine $G$ -stable chart $X_{σ}$ of steps 1.2 and 2.1; the charts therefore cover $X^{s s} (L)$ .

Remarks

Why the power is needed. The affineness conclusion is obtained through the very ample power $L^{\otimes m}$ of [F1]; the section $σ$ itself is replaced by its power, which does not change the nonvanishing locus by step 1.1.
No separatedness hypothesis. The argument uses only the closed-immersion presentation of a projective variety and the standard affine charts of projective space.

Depends on

Used by

Dependency tree · two levels

118 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

Michel Brion, Introduction to actions of algebraic groups, Les cours du CIRM 1 (2010), no. 1, 1-22 (standard reference, not scraped)
Victoria Hoskins, Moduli Problems and Geometric Invariant Theory, FU Berlin lecture notes (2015/16) (standard reference, not scraped)
I. Dolgachev, Lectures on Invariant Theory, London Mathematical Society Lecture Note Series 296, Cambridge University Press, 2003 (standard reference, not scraped)