[Paper Review] Separable rational connectedness and weak approximation in positive characteristic

Jason Starr, Zhiyu Tian|arXiv (Cornell University)|Jul 16, 2019

Algebraic Geometry and Number Theory6 references4 citations

TL;DR

This paper establishes a cohomological criterion for separable rational connectedness in smooth projective varieties of Picard number one in positive characteristic: a variety is separably rationally connected if and only if its Hodge cohomology groups $ H^0(X, \Omega^i_X) $ vanish for all $ i = 1, \dots, \dim X $. The result is applied to prove weak approximation for Fano complete intersections under favorable characteristic and degree conditions.

ABSTRACT

In this short note we give a characterization of smooth projective varieties of Picard number one that are separably uniruled but not separably rationally connected. We also give a sufficient condition involving the torsion order and the uniruling index for a smooth Fano variety of Picard number one to be separably rationally connected. As an application, we prove some weak approximation results for Fano complete intersections in positive charactersitic. For example, we show that weak approximation holds at place of strong potentially good reduction for a Fano complete intersection in $\mathbb{P}^n$ of type $(d_1, \ldots, d_c)$ in characteristic $p$ such that $n>d_1+\ldots +d_c, p>d_1, \ldots, d_c.$

Motivation & Objective

• To characterize smooth projective varieties of Picard number one that are separably uniruled but not separably rationally connected.
• To establish a sufficient condition for separable rational connectedness in Fano varieties of Picard number one using torsion order and uniruling index.
• To prove weak approximation results for Fano complete intersections in positive characteristic under degree and characteristic constraints.
• To generalize and improve prior results on separable rational connectedness and weak approximation from [Tia15] and [STZ18].

Proposed method

• Uses the notion of $ r $-uniruling and defines the $ r $-uniruling index as the GCD of degrees of generically finite $ r $-fold fiber product morphisms.
• Applies the concept of separable uniruling and separable rational connectedness via separable evaluation morphisms on fiber products.
• Employs the vanishing of $ H^0(X, \Omega^i_X) $ for $ i = 1, \dots, \dim X $ as a key cohomological criterion for separable rational connectedness.
• Leverages the structure of free rational curves and their pullbacks to the cotangent bundle to define the subsheaf $ \mathcal{D} \subset \Omega_X $.
• Uses the reflexive sheaf $ (\Lambda^{n-r}\mathcal{D})^{**} $ and its triviality due to Picard number one and degree zero to construct a global section of $ \Omega^{n-r}_X $.
• Applies results from [CL16] on torsion order and uniruling index to deduce separable rational connectedness of central fibers in families over DVRs.

Experimental results

Research questions

• RQ1 Under what conditions is a smooth projective variety of Picard number one that is separably uniruled but not separably rationally connected?
• RQ2 What is the precise cohomological criterion that guarantees separable rational connectedness in positive characteristic?
• RQ3 How do torsion order and uniruling index determine separable rational connectedness in Fano varieties of Picard number one?
• RQ4 Does weak approximation hold for Fano complete intersections in positive characteristic under degree and characteristic bounds?
• RQ5 Can the assumption of (strong) potentially good reduction in weak approximation theorems be relaxed using new criteria on central fibers?

Key findings

• A smooth projective variety $ X $ of Picard number one over an algebraically closed field of positive characteristic is separably rationally connected if and only if $ H^0(X, \Omega^i_X) = 0 $ for all $ i = 1, \dots, \dim X $.
• For a Fano complete intersection $ X \subset \mathbb{P}^n $ of type $ (d_1, \dots, d_c) $ with $ n > \sum d_i $ and $ p > \max(d_1, \dots, d_c) $, the central fiber over a DVR of residue characteristic $ p $ is separably rationally connected.
• The uniruling index of such a complete intersection divides $ \prod_{i=1}^c (d_i!) $, and the torsion order also divides this product.
• Weak approximation holds at every place of (strong) potentially good reduction for smooth families of complete intersections in $ \mathbb{P}^n $ of Fano index $ \geq 2 $, provided $ p > \max(d_1, \dots, d_c) $.
• The proof of weak approximation relies on the central fiber being separably rationally connected, which follows from Corollary 9, thus improving [STZ18, Theorem 1.1] by relaxing the reduction assumption.
• For complete intersections in homogeneous spaces, an explicit criterion for separable rational connectedness is given using the Fano index and the product of factorials of intersection numbers with free rational curves, under characteristic bounds.