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The selected zero fibers are finite etale schemes

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

Statement

For every point $a \in T^{\circ}$ , the zero scheme $S_{a}$ of the universal functional after base change to $L = κ (a)$ is a finite étale $L$ -scheme, possibly empty. The same holds after every extension of $L$ , so it is geometrically smooth of dimension zero. In particular for a chosen $a \in T^{\circ} (k)$ it is a finite disjoint union of copies of $Spec k$ , all in $X_{s m / k}$ . This also identifies the generic fiber over $k (T)$ : it is finite étale, not merely a set of expected dimension. No dominance is needed or asserted in this argument.

Facts & Assumptions

Given: The nonempty parameter open and the actual zero scheme, which is closed in the affine finite-type scheme $X_{L}$ .

Goal: Prove geometric smoothness and finiteness with the correct scheme structure.

[F1]

One nonempty parameter open gives smooth-locus avoidance and transversality makes the zero scheme lie in $W_{L}$ with derivative determinant invertible at every point, and retains this after field extension. Finite-dimensional families of generated bundle functionals supplies the scheme-theoretic fiber description.

[F2]

External finite-type dimension, rational-point and etale-fiber inputs supplies: a finite-type algebra over a field with zero differential module is étale, and an étale finite-type algebra over a field is a finite product of finite separable field extensions. These statements have no characteristic-zero restriction.

Proof

1.1F1

Locally on $W_{L}$ trivialize the rank-three bundle and write the section coefficients $g_{1}, g_{2}, g_{3}$ . On their zero scheme the conormal sequence gives a right-exact map $O_{S_{a}}^{3} \overset{(d g_{1}, d g_{2}, d g_{3})}{\to} Ω_{W_{L} / L}^{1} ∣_{S_{a}} ⟶ Ω_{S_{a} / L}^{1} ⟶ 0.$ The middle bundle has rank three, and the first map has invertible determinant by [F1]. It is an isomorphism, so $Ω_{S_{a} / L}^{1} = 0$ . The frame-independent determinant condition guarantees that this conclusion glues across charts.

2.1F1F2step 1.1∎

As a closed subscheme of affine $X_{L}$ , write $S_{a} = Spec B_{a}$ with $B_{a}$ a finite-type $L$ -algebra. Step1.1 gives $Ω_{B_{a} / L} = 0$ , so [F2] makes $B_{a}$ a finite product of finite separable extensions of $L$ . This is finite étale; the empty case is the zero algebra and empty product. After any extension these remain finite étale. For $L = k$ algebraically closed, every factor is $k$ itself. Apply [F1] to see all these points lie in the original smooth locus. The same argument applies to the generic point of $T^{\circ}$ and proves the claimed generic-fiber statement.

Depends on

Used by · 1

Bertini functional with finite smooth zero locus

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