[Paper Review] Grothendieck-Serre Conjecture I: Appendix

Ivan Panin, Anastasia Stavrova+1 more|ArXiv.org|Oct 28, 2009

Algebraic Geometry and Number Theory5 references5 citations

TL;DR

This appendix establishes a Bertini-type theorem for transverse intersections of hypersurfaces with smooth subvarieties over infinite fields, extending Artin's results on Artin neighborhoods. It proves that for sufficiently general hypersurfaces of high enough degree containing finitely many points, the intersection with a smooth, pure-dimensional subvariety is transverse and preserves irreducibility, enabling key applications in the Grothendieck–Serre conjecture via cohomological descent and Hilbert's Theorem 90 for Azumaya algebras.

ABSTRACT

We prove here some supplementary statements that appeared without proof in I. Panin, A. Stavrova, N. Vavilov, On Grothendieck--Serre's conjecture concerning principal $G$-bundles over reductive group schemes:I, arXiv:0905.1418

Motivation & Objective

• To extend Artin’s Bertini-type theorem to the case of smooth subvarieties over infinite fields with prescribed points.
• To establish conditions under which intersections of general hypersurfaces with a smooth subvariety are transverse.
• To prove that such intersections preserve irreducibility (or geometric irreducibility) when the codimension is less than the dimension.
• To support the proof of the Grothendieck–Serre conjecture by constructing suitable neighborhoods and applying cohomological descent.
• To utilize the Shapiro–Faddeev lemma and Hilbert’s Theorem 90 for Azumaya algebras to trivialize bundles over semi-local schemes.

Proposed method

• Use the blow-up of projective space at a finite set of points to construct a very ample line bundle on the blow-up, enabling a closed embedding into a higher-dimensional projective space.
• Identify global sections of the ideal sheaf of degree-d hypersurfaces vanishing at given points with hyperplanes in the embedded space.
• Apply Artin’s transversality result (Theorem 2.1 of [A, Exp. XI]) to the embedded variety to deduce transversality of intersections.
• Enlarge the degree d to ensure transversality at points lying over the base points, using the surjectivity of global sections of O(d)/I^2(d).
• Use the Weil restriction and Shapiro–Faddeev lemma to relate cohomology classes over P^1 to those over a base change via a finite étale cover.
• Apply Hilbert’s Theorem 90 for Azumaya algebras to trivialize principal bundles over semi-local schemes, leveraging the triviality of bundles over fibers and Horrocks’ theorem on vector bundles.

Experimental results

Research questions

• RQ1 Under what conditions does a general family of hypersurfaces of sufficiently high degree intersect a smooth subvariety transversally?
• RQ2 When does such an intersection preserve the irreducibility (or geometric irreducibility) of the subvariety?
• RQ3 How can one construct a suitable Artin neighborhood for a smooth scheme over an infinite field with finitely many points removed?
• RQ4 Can cohomological descent techniques combined with Hilbert’s Theorem 90 trivialize principal bundles over P^1 with values in reductive group schemes?
• RQ5 What role does the Weil restriction play in lifting cohomology classes from a base to a finite cover in the context of the Grothendieck–Serre conjecture?

Key findings

• For any infinite field k, a finite set of closed points in P^n_k, and a smooth, pure-dimensional subvariety V' ⊂ V ⊂ P^n_k, there exists an integer d such that for sufficiently general hypersurfaces of degree d containing the points, the intersection with V' is transverse.
• If V is irreducible (resp. geometrically irreducible) and s < r = dim V, then for the same d and general hypersurfaces, the intersection Y ∩ V is irreducible (resp. geometrically irreducible).
• The construction of the embedding via blow-up and very ampleness ensures that transversality over the algebraic closure implies transversality over the base field.
• The bundle F over P^1 ×_U V is trivial because it is trivial on fibers and the associated vector bundle is trivial by Horrocks’ theorem.
• The triviality of the Azumaya algebra cohomology class over the semi-local base V implies that the H-bundle F is pulled back from V, hence trivial.
• The original G-bundle E over P^1 is pulled back from a G-bundle over the base U, proving the key descent step in the Grothendieck–Serre conjecture.