[Paper Review] Families of rationally simply connected varieties over surfaces and torsors for semisimple groups

A. J. de Jong, Xuhua He+1 more|ArXiv.org|Sep 30, 2008

Algebraic Geometry and Number Theory19 references4 citations

TL;DR

This paper establishes that a family of projective homogeneous varieties $G/P$ over the function field of a surface over an algebraically closed field of characteristic zero has a rational point if certain cohomological conditions are satisfied. Using an algebro-geometric analogue of simple connectedness via the projective line, the authors prove Serre’s Conjecture II in Galois cohomology for such fields, completing the conjecture in this setting by showing the only obstruction is a Brauer class on the base surface.

ABSTRACT

Under suitable hypotheses, we prove that a form of a projective homogeneous variety $G/P$ defined over the function field of a surface over an algebraically closed field has a rational point. The method uses an algebro-geometric analogue of simple connectedness replacing the unit interval by the projective line. As a consequence, we complete the proof of Serre's Conjecture II in Galois cohomology for function fields over an algebraically closed field.

Motivation & Objective

• To establish the existence of rational sections for families of projective homogeneous varieties $G/P$ over surfaces, under cohomological conditions.
• To complete the proof of Serre’s Conjecture II in Galois cohomology for function fields of surfaces over algebraically closed fields.
• To develop and apply the theory of rationally simply connected varieties via chains of free lines and very twisting surfaces.
• To show that the only obstruction to rational sections is a Brauer class on the base surface.
• To provide a geometric framework analogous to topological simple connectedness, replacing the unit interval with the projective line.

Proposed method

• Introduce the notion of rational simple connectedness via chains of free lines, formalized through Hypothesis 6.8.
• Define and utilize 'very twisting surfaces'—morphisms $f: \mathbb{P}^1 \to \overline{\mathcal{M}}_{0,1}(Y,1)$ with ample pullbacks of tangent bundles of evaluation and forgetful maps.
• Use Kontsevich stable maps and stable sections to analyze moduli spaces of rational curves on varieties.
• Construct 'peaceful chains' and 'porcupines' as geometric tools to control rational curves and their degenerations.
• Apply limit techniques and specialization to reduce the problem to very general complete intersections on the base surface.
• Leverage the existence of very twisting surfaces in Fano complete intersections under degree conditions $\sum d_i^2 \leq n$ to verify the key hypotheses.

Experimental results

Research questions

• RQ1 Under what conditions does a family of projective homogeneous varieties over a surface admit a rational section?
• RQ2 Can the notion of rational simple connectedness be formalized in algebraic geometry to generalize the GHS theorem from curves to surfaces?
• RQ3 Is the Brauer class the only obstruction to rational sections in families over surfaces, as suggested by Serre’s Conjecture II?
• RQ4 Do very twisting surfaces exist on general complete intersections of type $(d_1,\dots,d_c)$ in $\mathbb{P}^n$ when $\sum d_i^2 \leq n$?
• RQ5 Can the theory of stable maps and moduli spaces of curves be used to prove rational simple connectedness in higher-dimensional families?

Key findings

• A family of projective homogeneous varieties $G/P$ over the function field of a surface has a rational point if the base satisfies a cohomological condition involving the Brauer group and the variety is rationally simply connected by chains of free lines.
• The main result (Corollary 12.2) confirms Serre’s Conjecture II in Galois cohomology for function fields of surfaces over algebraically closed fields of characteristic zero.
• A very general complete intersection $Y$ of type $(d_1,\dots,d_c)$ in $\mathbb{P}^n$ is rationally simply connected by chains of free lines if $\sum d_i^2 \leq n$, as shown by the rational connectedness of spaces of lines and pairs of lines through general points.
• The existence of a very twisting surface on such a $Y$ is sufficient to imply the main theorem, and such surfaces exist under the condition $\sum d_i^2 \leq n$, though explicit construction is only complete for hypersurfaces ($c=1$).
• The paper provides a geometric proof of Tsen’s theorem for families of complete intersections in characteristic zero using the theory of very twisting surfaces and rational simple connectedness.
• The method establishes that the only obstruction to rational sections is a Brauer class on the base surface, confirming the principle that 'the only obstruction is a Brauer class' in this context.