[Paper Review] Log rationally connected surfaces

Yi Zhu|arXiv (Cornell University)|Dec 8, 2014

Algebraic Geometry and Number Theory10 references3 citations

TL;DR

This paper establishes a numerical criterion for log rationally connected log surface pairs over the complex numbers, proving that such pairs are characterized by the vanishing of all symmetric powers of the log cotangent bundle and negative log Kodaira dimension. The key result generalizes Castelnuovo's rationality criterion to the log setting via the log minimal model program and a novel link to strong approximation over function fields of curves.

ABSTRACT

In this paper, combining the works of Miyanishi-Tsunoda and Keel-McKernan, we prove the log Castelnuovo's rationality criterion for smooth quasiprojective surfaces over complex numbers.

Motivation & Objective

• To establish a numerical characterization of log rationally connected log surface pairs over the complex numbers.
• To generalize Castelnuovo's classical rationality criterion to the logarithmic setting.
• To clarify the relationship between the non-existence of pluri log one-forms and the strong approximation problem over function fields of curves.
• To resolve the log ruled case in the log minimal model program by connecting it to integral sections of affine line fibrations.
• To demonstrate that negative log Kodaira dimension and finite fundamental group are insufficient to guarantee log rational connectedness.

Proposed method

• Combines the log minimal model program (log MMP) with classification results from Miyanishi-Tsunoda and Keel-McKernan to reduce to known end cases.
• Uses the fact that log rational connectedness is preserved under divisorial contractions and deletion of closed subsets of codimension one.
• Applies a base change argument via finite covers of P^1 to construct integral sections over function fields of curves.
• Employs explicit analysis of pullbacks of log one-forms to show non-vanishing in certain cases, contradicting the vanishing condition.
• Leverages strong approximation results for A^1 and P^1 over function fields of curves to construct rational curves connecting general points.
• Uses Van Kampen’s theorem and log Albanese variety theory to compute log irregularity and fundamental group in the counterexample.

Experimental results

Research questions

• RQ1 What numerical conditions characterize log rationally connected log surface pairs over C?
• RQ2 How does the vanishing of symmetric powers of the log cotangent bundle relate to log rational connectedness?
• RQ3 Can the log ruled case in the log MMP be proven to be log rationally connected using strong approximation over function fields?
• RQ4 Is the condition H^0(X, S^12 Ω^1_X(log D)) = 0 sharp in the log Castelnuovo criterion?
• RQ5 Can negative log Kodaira dimension and trivial fundamental group coexist with non-log-rational connectedness?

Key findings

• The log Castelnuovo criterion holds: a log smooth log surface pair (X,D) is log rationally connected if and only if H^0(X, (Ω^1_X(log D))^⊗m) = 0 for all m ≥ 1.
• The condition H^0(X, S^12 Ω^1_X(log D)) = 0 is sharp, as shown by the failure of the criterion with lower symmetric powers.
• The log ruled case is proven to be log rationally connected by constructing integral sections via base change and strong approximation over function fields.
• A counterexample exists where π₁(X−D) is trivial, q(X,D) = 0, and κ(X,D) = −∞, yet (X,D) is not log rationally connected.
• Log rational connectedness implies finite fundamental group, but this condition is not sufficient, as shown by the counterexample.
• The log analogue of Campana–Kollár–Miyaoka–Mori’s rational connectedness theorem fails for log surface pairs, as demonstrated by the log Fano pair (P², {xy=0}).