Batyrev–Manin conjecture for singular degree-one del Pezzo surfaces

Let $A,B\in\mathbb{Z}$ satisfy $4A^3-27B^2\neq 0$, let $Q(u,v)$ be a positive-definite binary quadratic form, and define

$$S_{A,B,Q}:=\{y^2=x^3+AxQ(u,v)^2+BQ(u,v)^3\}\subset\mathbb{P}(2,3,1,1).$$

For rational points represented by integral $(x,y,u,v)$ with $\gcd(x,u,v)=1$, use the height

$$\tilde{h}((x,y,u,v))=\max(|x|^{1/2},|y|^{1/3},|u|,|v|).$$

Let $U_1=\mathbb{P}(2,3,1,1)\setminus\{y=0\}$ and $U_2=\mathbb{P}(2,3,1,1)\setminus\{Q(u,v)=0\}$, and define

$$\tilde{N}(T):=\#\{(x,y,u,v)\in S_{A,B,Q}(\mathbb{Z})\cap U_1\cap U_2:\tilde{h}((x,y,u,v))\leq T,\ \gcd(x,u,v)=1\}.$$

Assume that $S_{A,B,Q}(\mathbb{Q})\neq\emptyset$. Batyrev–Manin conjecture. There exists a constant $C_{A,B,Q}$ such that

$$\tilde{N}(T)\sim C_{A,B,Q}T\log(T)^{\varrho_{A,B,Q}-1},$$

where $\varrho_{A,B,Q}$ is the rank of the Picard group over $\mathbb{Q}$ of the desingularization $\widetilde{S}_{A,B,Q}$ of $S_{A,B,Q}$. This is a refinement of Manin's conjecture for these singular degree-one del Pezzo surfaces; the constant is related to Peyre's constant and need not be nonzero. Asymptotics are not known for any del Pezzo surface of degree one, although upper and lower bounds have been studied.