[Paper Review] A note on hyperbolicity for log canonical pairs

Roberto Svaldi|arXiv (Cornell University)|Oct 9, 2014

Algebraic Geometry and Number Theory6 references3 citations

TL;DR

This paper establishes conditions under which the log canonical divisor $K_X + \Delta$ is nef or ample for a log canonical pair $(X, \Delta)$, by assuming the absence of non-constant rational curves in the open strata of the non-klt locus. It generalizes the Cone Theorem and provides a Nakai-type criterion for ampleness, extending results to arbitrary singularities in partial cases.

ABSTRACT

Given a log canonical pair $(X, \Delta)$, we prove that $K_X+\Delta$ is nef assuming there is no non constant map from the projective line with values in the open strata of the stratification induced by the non klt locus of $\Delta$. This implies a generalization of the Cone Theorem. Moreover, we give a criterion of Nakai type to determine when under the above condition $K_X+\Delta$ is ample. We prove some partial results in the case of arbitrary singularities.

Motivation & Objective

• To determine conditions under which $K_X + \Delta$ is nef for a log canonical pair $(X, \Delta)$.
• To generalize the Cone Theorem in the context of log canonical pairs with controlled rational curves.
• To establish a Nakai-type criterion for ampleness of $K_X + \Delta$ under the same rational curve condition.
• To extend results to cases with arbitrary singularities, providing partial characterizations.

Proposed method

• Assumes no non-constant map from $\mathbb{P}^1$ to the open strata of the stratification induced by the non-klt locus of $\Delta$.
• Applies techniques from birational geometry and log minimal model theory to analyze the nefness of $K_X + \Delta$.
• Uses the structure of the non-klt locus to define a stratification and analyze rational curves in its open strata.
• Employs a Nakai-type criterion involving intersection numbers to determine ampleness of $K_X + \Delta$.
• Considers the behavior of $K_X + \Delta$ under modifications and restricts to cases with controlled singularities.

Experimental results

Research questions

• RQ1 Under what conditions is $K_X + \Delta$ nef for a log canonical pair $(X, \Delta)$?
• RQ2 How can the Cone Theorem be generalized in the context of log canonical pairs?
• RQ3 What Nakai-type criterion ensures that $K_X + \Delta$ is ample under the rational curve condition?
• RQ4 To what extent do the results extend to pairs with arbitrary singularities?

Key findings

• If there are no non-constant rational curves in the open strata of the non-klt locus, then $K_X + \Delta$ is nef.
• The absence of such rational curves implies a generalization of the Cone Theorem for log canonical pairs.
• A Nakai-type criterion is established to determine when $K_X + \Delta$ is ample under the same condition.
• Partial results are obtained for pairs with arbitrary singularities, extending the framework beyond the log canonical case.