Abstract: |
Manin's conjecture predicts an asymptotic formula for the counting function of the number of rational points on a Fano variety, and it has an explicit asymptotic formula in terms of geometric invariants of the underlying variety. The original conjecture, which predicts an asymptotic formula after removing a closed exceptional set, is wrong due to covering families of subvarieties violating the compatibility of Manin's conjecture, and its refinement suggested by Peyre removes a thin set instead of a closed set. In this talk, I will discuss that subvarieties violating the compatibility of Manin's conjecture only forms a thin set using the minimal model program and the boundedness of log Fano varieties. Then I will discuss our conjecture on the birational finiteness of generically finite covers violating the compatibility of Manin's conjecture. This is joint work with Brian Lehmann. |