The cone of moving curves of a smooth Fano three- or fourfold

TL;DR

This paper characterizes the cone of moving curves in smooth Fano three- and fourfolds using finitely many equations derived from divisorial contractions and nef divisors on birational models, advancing understanding of their geometric structure.

Contribution

It provides a finite description of the cone of moving curves for certain Fano varieties, linking it to divisorial contractions and nef divisors on birational models.

Findings

Finite equations describe the cone of moving curves.

Equations are induced by exceptional divisors and nef divisors.

Results contribute to the classification of Fano varieties.

Abstract

We describe the closed cone of moving curves of smooth Fano three- and fourfolds by giving finitely many equations that cut out the cone. The equations are induced by the exceptional divisors of divisorial contractions and by nef divisors on birational models obtained by sequences of flips.The results presented here are part of the author's Ph.D. thesis, written under the supervision of Stefan Kebekus.