[Paper Review] Zariski decomposition of curves on algebraic varieties

Brian Lehmann, Jian Xiao|arXiv (Cornell University)|Jul 15, 2015

Geometry and complex manifolds43 references3 citations

TL;DR

This paper introduces a Zariski decomposition for curve classes on algebraic varieties, enabling a refined theory of the volume function for curves. Using this decomposition, the authors establish fundamental positivity results, including a Morse-type inequality, and reveal surprising connections between curve volume and mobility, leading to applications in birational geometry and a refined structure theorem for the movable cone of curves.

ABSTRACT

We introduce a Zariski decomposition for curve classes and use it to develop the theory of the volume function for curves defined by the second named author. For toric varieties and for hyperk\ahler manifolds the Zariski decomposition admits an interesting geometric interpretation. With the decomposition, we prove some fundamental positivity results for curve classes, such as a Morse-type inequality. We compare the volume of a curve class with its mobility, yielding some surprising results about asymptotic point counts. Finally, we give a number of applications to birational geometry, including a refined structure theorem for the movable cone of curves.

Motivation & Objective

• To develop a Zariski decomposition theory for curve classes on algebraic varieties.
• To extend the volume function for curves beyond its original definition using this decomposition.
• To establish new positivity results, such as a Morse-type inequality, for curve classes.
• To explore geometric interpretations of the decomposition in special cases like toric varieties and hyperkähler manifolds.
• To connect the volume of a curve class with its mobility, yielding insights into asymptotic point counts.

Proposed method

• Introduce a Zariski decomposition for curve classes analogous to the classical decomposition for divisors.
• Use the decomposition to define and analyze the volume function for curves via limit processes of linear systems.
• Apply the decomposition to toric varieties and hyperkähler manifolds to reveal geometric interpretations of the components.
• Derive a Morse-type inequality by analyzing the positivity properties of the nef and negative parts in the decomposition.
• Compare the volume of a curve class with its mobility, using asymptotic counting of points on curves.
• Use the decomposition to refine the structure of the movable cone of curves in birational geometry.

Experimental results

Research questions

• RQ1 How can a Zariski decomposition be meaningfully defined for curve classes, given the lack of a divisorial analogue in the curve setting?
• RQ2 What geometric insights does the Zariski decomposition for curves provide in special varieties like toric or hyperkähler manifolds?
• RQ3 How does the volume of a curve class relate to its mobility in asymptotic point counts?
• RQ4 What new positivity properties, such as a Morse-type inequality, can be derived from the decomposition?
• RQ5 How does the decomposition refine the structure of the movable cone of curves in birational geometry?

Key findings

• The Zariski decomposition for curve classes provides a canonical decomposition into a nef and a negative part, enabling a volume function with strong positivity properties.
• A Morse-type inequality is established for curve classes, generalizing classical inequalities in the context of curve positivity.
• The volume of a curve class is shown to be related to its mobility, leading to unexpected asymptotic behaviors in point counting.
• In toric varieties and hyperkähler manifolds, the Zariski decomposition admits a natural geometric interpretation tied to the underlying geometry.
• The decomposition yields a refined structure theorem for the movable cone of curves, enhancing understanding of birational geometry.
• The results reveal that volume and mobility are not directly comparable in the asymptotic regime, challenging prior expectations.