The cone of moving curves of a smooth Fano-threefold
Sammy Barkowski
TL;DR
This paper proves that the cone of moving curves in a smooth Fano-threefold is polyhedral, using duality with pseudoeffective divisors and relating extremal rays via Mori contractions.
Contribution
It establishes the polyhedrality of the cone of moving curves for smooth Fano-threefolds, connecting it with the dual cone of pseudoeffective divisors and extremal rays on birational models.
Findings
The cone of moving curves is polyhedral.
Duality with the cone of pseudoeffective divisors is used.
Extremal rays correspond to Mori contractions.
Abstract
In this short note we show that the closed cone of moving curves of a smooth Fano-threefold is polyhedral. The proof relies on a famous result of Bucksom, Demailly, Paun and Peternell which says that the strongly movable cone is dual to the cone of pseudoeffective divisors. Finally, we relate the extremal rays of the cone of moving curves on a threefold with extremal rays in the cone of strongly movable curves on a birational model obtained by a Mori contraction.
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Taxonomy
TopicsGeometry and complex manifolds · Algebraic Geometry and Number Theory · Geometric Analysis and Curvature Flows