[Paper Review] Foliations and Rational Connectedness in Positive Characteristic

Mingmin Shen|ArXiv.org|Jun 4, 2009

Algebraic Geometry and Number Theory5 references3 citations

TL;DR

This paper introduces the notion of freely rationally connected (FRC) varieties in positive characteristic and proves that such varieties admit a finite purely inseparable morphism to a separably rationally connected (SRC) variety. Using foliation theory and formal neighborhood analysis, the authors establish a generalized Graber-Harris-Starr theorem and show that FRC varieties are simply connected, resolving key differences between rational connectedness and separable rational connectedness in positive characteristic.

ABSTRACT

In this paper, the technique of foliations in characteristic $p$ is used to investigate the difference between rational connectedness and separable rational connectedness in positive characteristic. The notion of being freely rationally connected is defined; a variety is freely rationally connected if a general pair of points can be connected by a free rational curve. It is proved that a freely rationally connected variety admits a finite purely inseparable morphism to a separably rationally connected variety. As an application, a generalized Graber-Harris-Starr type theorem in positive characteristic is proved; namely, if a family of varieties over a smooth curve has the property that its geometric generic fiber is normal and freely rationally connected, then it has a rational section after some Frobenius twisting. We also show that a freely rationally connected variety is simply connected.

Motivation & Objective

• To clarify the distinction between rational connectedness and separable rational connectedness in positive characteristic.
• To define and study the new class of freely rationally connected (FRC) varieties.
• To prove that every FRC variety admits a finite purely inseparable morphism to an SRC variety.
• To establish a positive characteristic analog of the Graber-Harris-Starr theorem for FRC families.
• To show that FRC varieties are simply connected, extending results from characteristic zero.

Proposed method

• Construct a canonical foliation D ⊂ TX using free rational curves on X, which is closed under Lie brackets and p-th powers in positive characteristic.
• Define a quotient morphism X → Y by the foliation D, showing that Y inherits separable uniruledness or FRC structure from X.
• Analyze the formal neighborhood of a free rational curve to control the behavior of the quotient process when it does not terminate in SRC.
• Use the non-termination condition to show that global regular formal functions on the formal neighborhood form a power series ring.
• Apply the quotient construction iteratively to reduce an FRC variety to an SRC variety after finitely many steps.
• Leverage the resulting SRC quotient to prove the existence of rational sections after Frobenius twisting in families, via base change and known results from de Jong–Starr.

Experimental results

Research questions

• RQ1 Does freely rationally connected imply separably rationally connected in positive characteristic, and if not, what is the gap?
• RQ2 Can one construct a finite purely inseparable morphism from an FRC variety to an SRC variety?
• RQ3 What is the role of formal neighborhoods of free rational curves in controlling the quotient process?
• RQ4 Does a family of FRC varieties over a curve admit a rational section after Frobenius twisting?
• RQ5 Is a proper normal FRC variety in positive characteristic simply connected?

Key findings

• Every freely rationally connected variety over an algebraically closed field of positive characteristic admits a finite purely inseparable morphism to a separably rationally connected variety.
• The quotient process via foliations terminates in finitely many steps with an SRC variety, provided the original variety is FRC.
• A generalized Graber-Harris-Starr theorem holds in positive characteristic: if the geometric generic fiber of a family over a curve is normal and FRC, then it has a rational section after some Frobenius twist.
• The algebraic fundamental group of a proper normal FRC variety in positive characteristic is trivial, i.e., such varieties are simply connected.
• The formal neighborhood of a free rational curve on an FRC variety has a power series ring of global regular functions if the quotient process does not terminate in SRC.