Power-saving refinement of the Manin–Peyre conjecture

Let $V$ be an “almost Fano” variety in the sense of Peyre's Definition 3.1, with $V(\mathbb{Q})\neq\varnothing$, and let $H$ be an anticanonical height function on $V(\mathbb{Q})$. Write

$$N_{U,H}(B):=\#\{x\in U(\mathbb{Q})\mid H(x)\leq B\},\qquad \rho=\operatorname{rank}(\operatorname{Pic}(V)).$$

Refinement of the Manin–Peyre conjectures. There exist a Zariski open subset $U$ of $V$, a polynomial $P_{U,H}$ of degree $\rho$, and $\delta\in(0,1)$ such that, for $B\geq 1$,

$$N_{U,H}(B)=c_{H,V}B P_{U,H}(\log(B))+O\mathopen{}\mathclose\bgroup\originalleft(B^{1-\delta}\aftergroup\egroup\originalright),$$

where the leading coefficient of $P_{U,H}$ agrees with Peyre's prediction. This is a power-saving asymptotic refinement of Manin's point-counting conjecture. The paper's abstract states that it is proved for the varieties defined by $x_1y_2y_3+x_2y_1y_3+x_3y_1y_2=0$ in the indicated products of projective spaces; the refinement is not asserted as proved for every almost Fano variety.