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Finite-dimensional families of generated bundle functionals

PDF p. 15: Two Bertini applications

Statement

Let $k$ be a field, $U$ a $k$ -scheme, and $E$ a vector bundle of constant rank $r$ on $U$ . Write $E^{\lor} = H o m_{O_{U}} (E, O_{U})$ , and suppose that global sections $ϕ_{1}, \dots, ϕ_{m}$ generate $E^{\lor}$ , meaning that the morphism $q : O_{U}^{m} \to E^{\lor}$ sending the standard basis to these sections is surjective. Here $m$ and $r$ may be zero.

The affine parameter space $A_{k}^{m} = Spec k [T_{1}, \dots, T_{m}]$ carries a universal functional $Φ = \sum_{i} T_{i} ϕ_{i}$ on the pullback of $E$ to $U \times_{k} A_{k}^{m}$ . Its zero-incidence subscheme $I$ is the closed subscheme defined locally by the coefficients of $Φ$ in a frame of $E^{\lor}$ . For every point $u \in U$ , evaluation gives a surjective linear map $κ (u)^{m} \to E^{\lor} \otimes_{O_{U}} κ (u)$ , and this remains surjective after any field extension of $κ (u)$ .

For every closed subscheme $Z ↪ U$ , the incidence subscheme constructed from the restricted sections on $Z$ is canonically $I \times_{U} Z$ , viewed as a closed subscheme of $Z \times_{k} A_{k}^{m}$ . More generally, for any $k$ -scheme $T$ and any $T$ -valued parameter $(a_{1}, \dots, a_{m})$ , the pullback of $I$ to $U \times_{k} T$ is the zero subscheme of the functional $\sum_{i} a_{i} ϕ_{i}$ .

Statement references

Kernels of surjections of vector bundles

Unimodular elements and trivial summands

Facts & Assumptions

Given: The field $k$ , the $k$ -scheme $U$ , the rank- $r$ vector bundle $E$ , and the surjective generating map $q$ in the statement.

Goal: Construct the universal zero-incidence subscheme, prove fiberwise surjectivity of evaluation, and identify its base changes.

[F1]

A vector bundle is a finite locally free module; its dual is the sheaf of homomorphisms to the structure sheaf, and a functional is a section of that dual. Unimodular elements and trivial summands

[F2]

A surjective morphism between finite locally free sheaves splits locally. Kernels of surjections of vector bundles

Proof

technique · Use local frames and the locally split generating map

1.1F1givenconstruct

Let $p : U \times_{k} A_{k}^{m} \to U$ be the projection. The coordinate functions $T_{i}$ and the pullbacks $p^{*} ϕ_{i}$ define a section $Φ = \sum_{i} T_{i} p^{*} ϕ_{i}$ of $p^{*} E^{\lor}$ , hence a functional $p^{*} E \to O_{U \times_{k} A_{k}^{m}}$ . On a trivializing open for $E$ , write its coefficients as $f_{1}, \dots, f_{r}$ . Define $I$ there by the ideal $(f_{1}, \dots, f_{r})$ . On overlapping trivializations the coefficient columns differ by an invertible matrix, so they generate the same ideal. These ideals glue to a closed subscheme $I$ ; equivalently, its ideal is the image of the functional $Φ$ . This also covers rank zero, where the ideal is zero.

2.1F2givenstep 1.1

By [F2], $q$ has a right inverse locally on $U$ . Pulling this local splitting back to the residue field $κ (u)$ of any point $u$ gives a right inverse to the evaluation map $κ (u)^{m} \to E^{\lor} \otimes κ (u)$ , so that map is surjective. Tensoring the same right inverse with any field extension of $κ (u)$ proves the asserted surjectivity over the extension.

2.2step 1.1givenalgebra

A morphism $T \to A_{k}^{m}$ is specified by the images $a_{i}$ of the coordinate functions. Pulling back the local coefficients $f_{j}$ along $U \times_{k} T \to U \times_{k} A_{k}^{m}$ substitutes $a_{i}$ for $T_{i}$ ; their generated ideal is therefore the image ideal of $\sum_{i} a_{i} ϕ_{i}$ . The scheme-theoretic inverse image of a closed subscheme is defined by the extended ideal, so this pullback is exactly the stated zero subscheme.

3.1F1step 1.1step 2.1step 2.2∎

For a closed subscheme $Z ↪ U$ , pulling the same coefficient ideals back to $Z \times_{k} A_{k}^{m}$ gives the coefficients of the restricted sections of $E^{\lor} ∣_{Z} ≅ (E ∣_{Z})^{\lor}$ in the restricted frames. Thus their incidence subscheme is canonically $I \times_{U} Z$ . Together with steps 1.1 through 2.2 this proves all the assertions.

Remarks

This construction alone does not assert smoothness, irreducibility, codimension, or avoidance for any parameter. Generation of $E$ does not by itself supply the assumed generation of $E^{\lor}$ .

Depends on

Used by · 7

Dependency tree · 3 levels

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