Geometric consistency of Manin's Conjecture

Abstract

We propose an explicit description of the exceptional set in Manin's Conjecture. Our proposal includes the rational point contributions from any subvariety or cover with larger geometric invariants. We prove that this set is contained in a thin subset of rational points, verifying there is no counterexample to Manin's Conjecture which arises from incompatibility of geometric invariants.the setis contained in a thin subset of X(F ).

An introduction to Geometric Manin's conjecture

“…In the late 1980s Yuri Manin and his collaborators (Victor Batyrev and Yuri Tschinkel) formulated a conjecture predicting an asymptotic formula for the counting function of rational points of bounded height on a smooth Fano variety defined over a number field ( [FMT89] and [BM90]), and this has been further revised over three decades by many mathematicians including but not limited to Emmanuel Peyre, Batyrev-Tschinkel, and Lehmann-Sengupta-Tanimoto in [Pey95], [BT98], [Pey03], [Pey17], and [LST18]. This conjecture has been formulated and summarized as Manin's conjecture or Batyrev-Manin's conjecture.…”

“…The assumption (1) has been verified over complex numbers in [LT19b, Theorem 1.1] using results on exceptional sets in Manin's conjecture ( [HJ17] and [LT17]) which are based on recent breakthroughs in the minimal model program ([BCHM10] and [Bir21]). The assumption (2) is wrong in general due to the presence of thin exceptional sets which are geometrically characterized in [LST18]. To overcome this situation, Brian Lehmann and the author proposed Geometric Manin's conjecture in [LT19b] as a replacement of the assumption (2), and this has been further revised in [BLRT20] by Roya Beheshti, Brian Lehmann, Eric Riedl, and the author.…”

Classifying sections of del Pezzo fibrations, II

“…Geometric Manin's Conjecture. Inspired by the thin set version of Manin's conjecture and the conjectural description of the exceptional set in [LST18], the authors proposed the first version of Geometric Manin's Conjecture in [LT19b]. The statement relies on the following invariant from the Minimal Model Program.…”

“…Roughly speaking Geometric Manin's Conjecture predicts two things. First, the conjecture predicts that there should be a "thin exceptional set" which can be described using the Fujita invariant as in [LST18].…”

“…Let π : X → B be a Fano fibration over a smooth projective curve. After removing all components of Sec(X /B) which factor through the exceptional set constructed by [LST18], there should be exactly |Br(X)| irreducible components of Sec(X /B) representing each sufficiently positive nef curve class which admits a relatively free section.…”