On the factor alpha in Peyre's constant (Q2871194)

Let \(S\) be a smooth del Pezzo surface of degree \(d\leq 7\), defined over a number field \(k\), and having infinitely many \(k\)-rational points. Let \(U\) be the complement of the lines on \(S\) and let \(H(\mathbf{x})\) be an anticannonical height function on \(S(k)\). Then Manin's conjecture, as refined by \textit{E. Peyre} [Duke Math. J. 79, No. 1, 101--218 (1995; Zbl 0901.14025)], predicts that NEWLINE\[NEWLINE\#\{\mathbf{x}\in U(k):\,H(\mathbf{x}) \leq B\}\sim \alpha(S)\beta(S)\omega(S,H)B(\log B)^{\rho(S)-1}NEWLINE\]NEWLINE as \(B\rightarrow\infty\), where \(\rho(S)\) is the rank of the Picard group \(\text{Pic}(S)\) over \(k\). Peyre defines \(\beta(S)\) and \(\omega(S,H)\) precisely, but this paper is concerned only with \(\alpha(S)\) which is essentially the volume of a certain polytope in \(\text{Pic}(S)\otimes_{\mathbb{Z}}\mathbb{R}\).NEWLINENEWLINEFor split del Pezzo surfaces the first author gave a formula for \(\alpha(S)\) in [\textit{U. Derenthal}, Math. Res. Lett. 14, No. 3, 481--489 (2007; Zbl 1131.14042)], while non-split surfaces of degree \(d\geq 5\) were handled by \textit{U. Derenthal} et al. [Algebra Number Theory 2, No. 2, 157--182 (2008; Zbl 1158.14032)].NEWLINENEWLINEThe principal theoretical result of the paper is that two smooth del Pezzo surfaces \(S\) and \(S'\) over \(k\) of the same degree, with \(S(k)\) and \(S'(k)\) non-empty and \(\rho(S)=\rho(S')\), have the same \(\alpha\)-constant whenever the Weil group \(W(S)\) is contained in a conjugate of \(W(S')\).NEWLINENEWLINEThis reduces the problem to a small number of cases. For degrees 3 and 4 the necessary computer calculations are described, leading to a complete classification with 17 and 9 cases respectively. As a representative example, it is shown that \(\alpha(S)=5/18\) for a smooth cubic surface with \(\rho(S)=4\), when \(S\) is isomorphic to \(\mathbb{P}^2\), blown up in two \(k\)-rational points and an orbit of size four.