The effective monoids of the blow-ups of Hirzebruch surfaces at points in general position
In the paper under review, the authors study the geometry of surfaces obtained as blow-ups of Hirzebruch surfaces along finite sets of points. Given a smooth projective rational surface \(S\) defined over an algebraically closed field \(k\) of arbitrary characteristic, we denote by \(\mathrm{Eff}(S)\) the set of all effective divisor classes on \(S\) modulo the numerical equivalence. Denote by \(\Sigma_{n}\) the \(n\)-th Hirzebruch surface. We say that a finite set of points in \(\Sigma_{n}\) is in general position if the curves of negative self-intersection on the surface obtained by blowing up \(\Sigma_{n}\) at such points are only \((-1)\)-curves and the \((-n)\)-th curve of the strict transform of the class of section \(C_{n}\) if \(n\) is greater than zero, and only \((-1)\)-curves in the remaining case. Let us denote by \(W_{n}^{r}\) the blow-up of the Hirzebruch surface \(\Sigma_{n}\) at \(r\) points in general position, where \(n\) and \(r\) are non-negative integers. The main result of the paper under review can be formulated as follows. Theorem A. The effective monoid \(\mathrm{Eff}(W_{n}^{r})\) is finitely generated if and only if one of the following holds: a) \(n\leq 2\) and \(r\leq 7\); b) \(n=3\) and \(r\leq 8\); or c) \(n\geq 4\) and \(r\leq n+4\). Moreover, the minimal generating sets for such monoids are completely determined. Let us recall that for a given smooth projective variety \(X\) the Cox ring is defined as \[\mathrm{Cox}(X) = \bigoplus_{(n_{1},\dots, n_{\ell})\in \mathbb{Z}^{\ell}} H^{0}(X, \mathcal{L}_{1}^{n_{1}} \otimes \dots \otimes \mathcal{L}_{\ell}^{n_{\ell}}),\] where \(\mathcal{L}_{1}, \dots, \mathcal{L}_{\ell}\) form a basis for the Picard group\(\mathrm{Pic}(X)\) of \(X\), and we also assume that linear and numerical equivalence of the group of Cartier divisors are the same. Regarding the finite generation problem of the Cox rings of surfaces \(W_{N}^{r}\), the authors deliver the following result. Theorem B. The Cox ring of \(W_{n}^{r}\) is finitely generated if and only if one of the following holds: a) \(n\leq 2\) and \(r\leq 7\); b) \(n=3\) and \(r\leq 8\); or c) \(n\geq 4\) and \(r\leq n+4\).
scientific article; zbMATH DE number 7343258
scientific article