Manin’s conjecture and the Fujita invariant of finite covers

Geometric consistency of Manin's conjecture

“…Theorem 1.4 generalizes earlier partial results in [BT98b, HTT15, LTT18, HJ17, LT17, Sen21]. These papers also establish some practical techniques for computing this thin set.…”

“…Since is adjoint rigid with respect to the divisor , [Sen21, Corollary 2.20] shows that for any generically finite cover of which has the same -value and which is adjoint rigid with respect to the pullback of the divisorial components of the branch locus are supported on the set . Furthermore, by [Sen21, Proposition 2.17] there is an upper bound on the degree of such covers depending only on , , and . Altogether there is a finite set of finite-index subgroups such that for some fiber of the corresponding étale cover has a projective closure which has the same -value as and is adjoint rigid.…”

“…The map is proper. Furthermore, [Sen21, Corollary 2.20] shows the map is the composition of a birational map with an étale map, so, in particular, it is étale outside of a subset in of codimension . We let denote the subgroup defined by the image of in .…”

Campana points, Height zeta functions, and log Manin's conjecture

“…It is well-documented in the case of rational points that in Conjecture 4.5 it is important to remove the contribution of a thin set Z from the counting function. There is a series of papers ([LT17], [Sen21], and [LST18]) studying birational geometry of thin exceptional subsets for rational points. In [LST18], Lehmann, Sengupta, and the author proposed a conjectural description of thin exceptional subsets and proved that it is indeed a thin set using the minimal model program and the boundedness of singular Fano varieties.…”

Moduli spaces of rational curves on Fano threefolds

“…In [Sen17] Sengupta shows that if X is a Fano variety then for any a-cover f : Y → X such that κ(K Y − f * K X ) = 0 the components of the branch divisor of f will have larger a-invariant than X. We will use explicit threefold birational geometry from [Kaw05] to prove a stronger restriction on the geometry of such branch divisors.…”