Batyrev–Popov conjecture on generators of del Pezzo Cox ring relations

Let $S_r$ be a del Pezzo surface of degree $(-K_{S_r})^2=9-r$ with $4\leq r\leq 8$. Let $G_r$ be a basis of the vector space

$$\bigoplus_{E:\,\text{$(-1)$-curve}} H^0(S_r,E)$$

for $r\leq 7$, and for $r=8$ let $G_8$ be a basis of

$$\left(\bigoplus_{E:\,\text{$(-1)$-curve}} H^0(S_8,E)\right)\oplus H^0(S_8,-K_{S_8}).$$

Write

$$\operatorname{Cox}(S_r)=k[G_r]/I_r,$$

where $I_r$ is the ideal of relations among these generators. Batyrev–Popov conjecture. The ideal $I_r$ is generated by quadrics. This conjecture concerns the defining relations of Cox rings of del Pezzo surfaces and is part of the study of their generators and syzygies. The source provides no resolution status, so it remains open in this record.