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The Geometric Manin's conjecture

The Geometric Manin's conjecture concerns the dimension and number of components of the space of rational curves on a projective variety. The aim of this seminar is to understand the paper Geometric Manin's conjecture and rational curves by Lehmann and Tanimoto. The starting points are Batyrev's heuristics and Manin's conjecture for rational points on Fano varieties over number fields. The authors observe that in order to have a reasonable control on the number of components, one must discount an exceptional set. They thus define the so-called Manin components as those that parametrize families of rational curves that capture the geometry of the variety, rather than the geometry of a subvariety or of a cover of. The Geometric Manin's Conjecture predicts an asymptotic formula for a generating function encoding the number and dimension of the Manin components of the space of morphisms of bounded degree from the projective line. The paper shows that the Geometric Manin's Conjecture holds for Fano threefolds of Picard rank 1 and index 2.

Everyone is welcome to attend. Upon request the talks can be streamed. If you wish to contribute a talk, contact Marta Pieropan: m.pieropan@uu.nl

Program

See the for the literature and a description of each talk.

Each talk lasts 2 hours including breaks.

1. Wednesday November 5 at 10:00 in room HFG 707 (Audrey Antoine)
   Introduction and the cone theorem
2. Tuesday November 11 at 10:00 in room HFG 707 (speaker TBA)
   The a- and b-invariants
3. Tuesday November 18 at 10:00 in room TBA (Soumya Sankar)
   Expected dimension
4. Tuesday November 25 at 10:00 in room HFG 707 (Justin Uhlemann)
   Number of components I
5. Tuesday December 2 at 10:00 in room HFG 707 (speaker TBA)
   Number of components II
6. Tuesday December 9 at 10:00 in room HFG 707 (Sara Mehidi)
   Geometric Manin Conjecture and examples
7. Tuesday December 16 at 10:00 in room HFG 707 (speaker TBA)
   Fano threefolds