On upper bounds of Manin type

The Manin–Peyre conjecture for smooth spherical Fano varieties of semisimple rank one

“…the proof of [55, Proposition 1.4] as the in Theorem 1.1 are blow-ups of toric varieties). Tanimoto [70, §7] proves the upper bounds for and .…”

Density of rational points on some quadric bundle threefolds

“…Manin [17] used height machinery to establish a lower bound supporting linear growth for all smooth Fano threefolds, possibly after an extension of the ground field. More recently, Tanimoto [26] has produced a range of upper bounds for various classes of Fano threefolds, but his work does not cover (1.1). The classification of Fano threefolds with Picard number 2 goes back to Mori and Mukai [18], but it is convenient to appeal to the summary of Iskovskikh and Prokhorov [15,Table 12.3].…”

Density of rational points near/on compact manifolds with certain curvature conditions

“…In particular, it also applies to pieces of projective varieties, that can in Monge form be represented by an l-nondegenerate function. For normal projective varieties, Tanimoto has recently given a very general upper bound for the number of rational points of bounded height, which is governed by a certain "δ invariant" that he introduces in [37]. His methods are of geometric nature and in particular require geometric understanding in the computation of the δ invariant.…”