Genericity can fail after enlarging the coefficient field
Where the issue occurs
Section 8.2, final sentence of the function-field discussion after the proof of Lemma 8.3 · p. 46
The same argument applies to a point after adjoining the other grid points to 𝓑, using Theorem 8.3.
This is the automatic-enlargement assertion being challenged, not the statement of Lemma 8.3. The quoted “Theorem 8.3” is the paper’s own wording.
Issue
At the end of Section 8.2, the paper asserts that a point generic over a coefficient field stays generic after adjoining other grid points. The grid lemma guarantees joint independence over the base field B₀, but not over an arbitrary larger field.
Counterexample
Let p and r be distinct grid vertices, with p on a curve or plane V over B₀ of dimension d ∈ {1, 2}. The selected coordinates are jointly algebraically independent over B₀; each vertex’s remaining coordinates are algebraic over B₀ and its own selected coordinates.
Choose selected coordinates a of p and b of r, and set B = B₀(a + b). Since b is transcendental over B₀(p), the point p remains generic over B, with transcendence degree d. But adjoining r gives:
One of p’s independent coordinates is now in the coefficient field. Thus p is no longer generic on V over B(r). The evaluation criterion that explicitly assumes genericity is unaffected.
Proposed repair by OSVerify
Restrict automatic preservation to adjoining other grid points to B₀. Their selected coordinates are jointly independent of the point’s selected coordinates, and adjoining their remaining curve coordinates is algebraic, so the point remains generic. For an arbitrary larger coefficient field, retain the explicit genericity hypothesis. The application over K_0 already establishes that hypothesis separately in the proof of Lemma 8.5; the later anchor applications use independence over B₀ and algebraic extensions. This correction resolves the identified overstatement without changing those arguments.
Related context
Background Section 8.2, Lemma 8.3 and Equation (8.5)
Gives joint genericity over B₀; it does not assert preservation over an arbitrary enlarged coefficient field.
Background Section 8.2, beginning of the same function-field paragraph
The preceding evaluation criterion explicitly assumes genericity over its coefficient field and is not challenged.
Downstream use Section 8.3, proof of Lemma 8.5, Equation (8.12)
Separately establishes the genericity needed over K₀; the report’s proposed correction preserves this application.
Downstream use Section 8.4, proof of Lemma 8.6, and Section 8.5, rank-condition argument
These anchor arguments use independence over B₀ and algebraic extensions. They explain the scope of the proposed correction, rather than identify additional errors.