The Batyrev–Tschinkel conjecture on the leading constant for point counts

Let $V$ be a smooth quasi-projective $K$-variety with an associated height function $H\colon V(K)\to\mathbb R$ coming from an adelic metrization on an ample line bundle $L$. Consider a natural projective completion $V\subseteq W$, fix a desingularisation $\rho\colon X\to W$, and set

$$M:=\rho^*\mathcal O_W(1).$$

For minimal closed subvarieties $V_\alpha\subseteq V$, define

$$\theta_\alpha:=\lim_{B\to\infty}\frac{N_{V_\alpha,K}(B)}{N_{V,K}(B)}>0.$$

Let $X_\alpha$ be the closure of $V_\alpha$ in $X$. Suppose that no proper closed subvariety $V'\subsetneq V$ contains all the $V_\alpha$, and that the restriction of $M$ to $X_\alpha$ is close to the anticanonical bundle in the sense specified by Batyrev and Tschinkel.

Batyrev–Tschinkel conjecture. Then

$$N_{V,K}(B)\sim\sum_\alpha c(X_\alpha)B^a\log^{b-1}B,$$

where $a$ and $b$ are the constants from the Batyrev–Manin conjecture and the constants $c(X_\alpha)$ are defined analogously to Peyre's constants.

The conjecture refines the expected asymptotic by describing the leading constant when the dominant contribution comes from several subvarieties. In the paper's application, the conjecture is resolved by passing to the regular locus $V\subseteq W$; the complement is lower-dimensional and does not affect the count.