A collection of \(r\leq 8\) points in \(\mathbb P^2\) is said to be in general position if no three are on a line, no six are on a conic and any cubic containing eight points is smooth at each of them. A Del Pezzo surface \(X_r=X_r(p_1, \ldots, p_r)\) is the blow up of \(\mathbb P^2\) at \(r\leq 8\) general points \(p_1\ldots, p_r\). For \(X_r\) one can choose the following basis of the Picard group \(\text{Pic}(X_r)\): the \(r\) line bundles \(\mathcal O[e_i]\) corresponding to the exceptional divisors \(e_i\) and \(\mathcal O[l]\) where \(l\) is the pullback of the line \(z=0\) in \(\mathbb P^2\). The Cox ring of \(X_r\) is defined by \[ \text{Cox}(X_r)=\underset{(m_0, \ldots, m_r)\in \mathbb Z^r}{\bigoplus} H^0(\mathcal O[m_0 l+m_1e_1+\cdots +m_rl_r]). \] \(\text{Cox}(X_r)\) is a \(\text{Pic}(X_r)\)–graded integral domain. For \(r<4\) the Cox ring is a polynomial ring because \(X_r\) is a toric variety [D. Cox, J. Algebr. Geom. 4, No. 1, 17–50 (1995; Zbl 0846.14032)].
Let \(E_r\) be the set of Picard classes of exceptional curves in \(X_r\) and \(k[E_r]\) be the \(\text{Pic}(X_r)\)–graded polynomial ring obtained by \(\deg([c])=[c]\). It is proved [V. Batyrev, O. Popov, in: Arithmetic of higher-dimensional algebraic varieties. Proc. workshop on rational and integral points of higher-dimensional varieties, Palo Alto, CA, USA, December 11–20, 2002. Prog. Math. 226, 85–103 (2004; Zbl 1075.14035)] that the map \(\Phi:k[E_r]\to\text{Cox}(X_r)\) sending each variable to the corresponding section is a \(\text{Pic}(X_r)\)–graded surjective homomorphism.
Let \(C_r=\text{Ker}(\Phi)\); then Batyrev and Popov conjectured that for \(4\leq r\leq 8\) \(C_r\) is generated by quadrics. In this paper the conjecture is proved for \(r=4,5\) and cubic surfaces without Eckart points. To prove the conjecture monomial group actions combined with Gröbner basis theory are used. The groups of symmetries of the configuration of the exceptional curves play a fundamental role. It is proved that although the action of the Weyl group on \(k[E_r]\) does not fix the ideal \(C_r\), it can be rediscovered as symmetries of the Gröbner fan of \(C_r\).