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Coordinate multiples of dual generators span all first jets

PDF p. 15: Theorem6.7: rank-three Bertini functional

Statement

Assume AC. Let $X = Spec A$ be finite type over an algebraically closed field, let $E$ be a rank-three bundle, and let $W ↪ X$ be a locally closed smooth $k$ -subscheme of pure dimension three. Choose $k$ -algebra generators $f_{1}, \dots, f_{d}$ of $A$ and generators $ϕ_{1}, \dots, ϕ_{m}$ of the coherent bundle $E$ . The finite indexed family $ϕ_{i}, f_{l} ϕ_{i} (1 \leq i \leq m, 1 \leq l \leq d)$ defines a parameter space $T = A_{k}^{N}$ , where $N = m (d + 1)$ ; linear relations among the listed sections are permitted. The evaluation $O_{X}^{N} ↠ E$ is surjective. On $W$ its first-jet map $j : O_{W}^{N} ↠ J^{1} (E ∣_{W})$ is also surjective, and $J^{1} (E ∣_{W})$ has rank twelve.

Here the first-jet bundle is defined by local pairs $(v, ω)$ in $E \oplus (Ω_{W / k}^{1} \otimes E)$ . Under a change of frame $G$ the pair transforms by $(v, ω) \mapsto (G v, G ω + (d G) v)$ . The product rule verifies the cocycle law and invertibility, so these charts define a vector bundle with exact sequence $0 \to Ω_{W / k}^{1} \otimes E \to J^{1} E \to E \to 0$ . The jet of a section with coefficient column $a$ is $(a, d a)$ . Thus at a closed point $x \in W$ the jet fiber is $(E ∣_{W})_{x} / m_{W, x}^{2} (E ∣_{W})_{x}$ over $k$ , where both the stalk and maximal ideal are on $W$ .

Facts & Assumptions

Given: The stated affine scheme and bundles. Generators exist by Finite global generation on an affine scheme applied to $E$ , with AC.

Goal: Obtain actual first-order separation, beyond fiberwise generation.

[F1]

The completed affine-generation and kernel lemmas give the finite generators and local splittings of surjections between bundles. Finite global generation on an affine scheme, Kernels of surjections of vector bundles, The Axiom of Choice

[F2]

Finite-type $k$ -schemes are Jacobson and their closed points are $k$ -rational. A coherent sheaf with no closed points in its support is zero. External finite-type dimension, rational-point and etale-fiber inputs

Proof

1.1F1givenconstruct

Evaluation is surjective because the listed sections already include the generators $ϕ_{i}$ . The transition formula in the Statement follows by differentiating $G a$ ; the product rule also shows that its composition for two frame changes equals the formula for their product. It therefore constructs $J^{1} E$ and the global $O_{W}$ -linear jet evaluation map from the indexed list. Since $W$ is smooth three-dimensional, $Ω_{W / k}^{1}$ has rank three; the exact sequence gives rank $3 + 9 = 12$ .

2.1F1F2step 1.1algebra

Fix a closed $x \in W$ . The elements $f_{l} - f_{l} (x)$ generate the maximal ideal of $x$ in $A$ , because $A$ is generated by the $f_{l}$ and its residue map is evaluation at their $k$ -values. The immersion induces a surjection $m_{X, x} / m_{X, x}^{2} ↠ m_{W, x} / m_{W, x}^{2}$ : locally a locally closed immersion is a quotient of the ambient local ring. Consequently the restrictions of those differences span the latter cotangent space. The values of the $ϕ_{i}$ span the three-dimensional bundle fiber. Constants times $ϕ_{i}$ can consequently realize an arbitrary prescribed value at $x$ . Each section $(f_{l} - f_{l} (x)) ϕ_{i}$ is in the $k$ -linear span of our indexed family, vanishes at $x$ , and has first-order class $d f_{l} \otimes ϕ_{i} (x)$ . These classes span the entire kernel $(m_{W, x} / m_{W, x}^{2}) \otimes_{k} E (x)$ of the jet-to-value map. Adjusting the initial chosen section by their linear combinations therefore realizes any prescribed first jet.

3.1F1F2step 2.1∎

The jet-evaluation cokernel is coherent. Step 2.1 makes its fiber zero at every closed point, and Nakayama's lemma makes its stalk zero there. Its support is closed and cannot be nonempty by the Jacobson property in [F2]. It is zero, so $j$ is a bundle surjection everywhere, not just on the closed points. It splits locally by [F1]. All constructions use ordinary products and derivatives, with no division by a positive integer.

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