The stacky Batyrev–Manin conjecture

Let $\mathcal X$ be a stack over a number field $F$, and write $\mathcal X\langle F\rangle$ for the set of $F$-isomorphism classes of $F$-points. Assume that $\mathcal X\langle F\rangle$ is Zariski-dense in $\mathcal X$. Let $(L,x)\in\operatorname{\overline{Eff}}_{\operatorname{orb}}(\mathcal X)$, where $\operatorname{\overline{Eff}}_{\operatorname{orb}}(\mathcal X)$ is the orbifold effective cone, and let $H:\mathcal X\langle F\rangle\to\mathbb R_{>0}$ be the corresponding height. Define

$$a(L,x):=\inf\{t\in\mathbb R\mid t\cdot(L,x)+K_{\mathcal X,\operatorname{orb}}\in\operatorname{\overline{Eff}}_{\operatorname{orb}}(\mathcal X)\}.$$

Let $b(L,x)$ be the codimension of the minimal face of $\operatorname{\overline{Eff}}_{\operatorname{orb}}(\mathcal X)$ containing $a(L,x)\cdot(L,x)+K_{\mathcal X,\operatorname{orb}}$.

Stacky Batyrev–Manin conjecture. There exists a thin set $T\subset\mathcal X\langle F\rangle$ and $C>0$ such that

$$\#\{\mathcal X\langle F\rangle-T\mid H(x)\leq B\}\sim_{B\to\infty}CB^{a(L,x)}\log(B)^{b(L,x)-1}.$$

This is the stack-theoretic analogue of the Batyrev–Manin prediction, with orbifold positivity and the canonical class governing the exponents. The paper notes that it does not additionally conjecture that $T$ is a union of images of breaking thin morphisms; the supplied text gives no resolution of the general conjecture.