English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

This survey paper discusses some of the recent progress in the study ofrational curves on algebraic varieties. It was written for the survey volume ofthe priority programme "Global Methods in Complex Geometry", supported by theDFG.  To start, we discuss existence results for rational curves on foliatedmanifolds. We construct the tangent morphism, define the variety of minimalrational tangents (VMRT) and name a number of results that show that many ofthe geometry properies of a uniruled variety are encoded in the projectivegeometry of the VMRT.  The results are then applied in two different settings. For one, we discussthe geometry of chains of rational curves on uniruled varieties, show how thelength of a variety can be determined from the VMRT and discuss the concreteexample of the moduli space of vector bundles on a curve.  On the other hand, the existence results for rational curves can be also beused to study manifolds which are known NOT to contain rational curves. Weemploy this method to study deformations of morphisms. Finally, we look atfamilies of varieties of canonically polarized manifolds over complex surfaces,and use non-existence results for rational curves on the base to relate thevariation of the family with the logarithmic Kodaira dimension of the base.

Existence of Rational Curves on Algebraic Varieties, Minimal Rational Tangents, and Applications