Rational curves on Fano threefolds

Abstract: Mori proved that a smooth Fano variety contains a lots of rational curves using famous Bend and Break technique. Thus it is natural to study the space of rational curves on a smooth Fano variety. Lines and conics on Fano threefolds are well studied, and one may ask what one can say about higher degree rational curves. Recently we established Movable Bend and Break for Fano threefolds claiming that any free curve of high degree breaks into the union of two free curves. A proof is intricate, and it relies on many properties of three dimensional MMP such as Mori’s classification of divisorial contractions on smooth projective threefolds. In this talk I would like to explain some aspects of our proof of Movable Bend and Break as well as an application to Batyrev’s conjecture predicting a polynomial growth of the number of components of bounded degree for the moduli space of rational curves. If time permits, then I also explain a relation of our study to Geometric Manin’s conjecture which is an inspiration of our study. This is joint work with Roya Beheshti, Brian Lehmann, and Eric Riedl.

algebraic geometry

Audience: researchers in the topic

ZAG (Zoom Algebraic Geometry) seminar

Series comments: Description: ZAG seminar

The seminar takes place on Tuesdays and Thursdays via Zoom. Zoom passwords are given via mailing list on Fridays. To join the mailing list go to the website.

If you use a calendar system, you can see the individual seminars at bit.ly/zag-seminar-calendar

Times vary to accommodate speakers time zones but times will be announced in GMT time.