Section rings of \(\mathbb{Q}\)-divisors on minimal rational surfaces (Q1720125)

Let us recall that for a Weil \(\mathbb{Q}\)-divisor \(D\) on a rational surface \(X\) the graded section ring of \(D\) is defined as \[ R(X,D) := \bigoplus_{\ell \geq 0}H^{0}(X, \lfloor \ell D \rfloor ). \] The main aim of the paper under review is to study section rings of divisors in the case of projective spaces \(\mathbb{P}^{m}\) and Hirzebruch surfaces \(\mathbb{F}_{n}\). {Theorem A}. Let \(D = \sum_{i=1}^{s} \alpha_{i} D_{i} \in \text{Div}(\mathbb{P}^{m}) \otimes_{\mathbb{Z}}\mathbb{Q}\) with \(\alpha_{i} = \frac{c_{i}}{k_{i}} \in \mathbb{Q}_{>0}\) written in reduced form and \(D_{i}\)'s are integral divisors. Then the section ring \(R(\mathbb{P}^{m},D)\) is generated in degree at most \(\text{max}_{0 \leq i \leq s}k_{i}\) with relations generated in degree at most \(2 \, \text{max}_{0 \leq i \leq s}k_{i}\) For a Hirzebruch surface \(\mathbb{F}_{n} = \text{Proj Sym}(\mathcal{O}_{\mathbb{P}^{1}} \oplus \mathcal{O}_{\mathbb{P}^{1}}(n))\), denote by \(u,v\) the projective coordinates on the base \(\mathbb{P}^{1}\) and let \(z,w\) be the projective coordinates on the fiber. {Theorem B}. Let \(D = \sum_{i=1}^{s} \alpha_{i} D_{i} \in \text{Div}(\mathbb{F}_{n}) \otimes_{\mathbb{Z}}\mathbb{Q}\) with \(\alpha_{i} = \frac{c_{i}}{k_{i}} \in \mathbb{Q}\) written in reduced form. Let \(V(f_{i}) = D_{i}\), where \(f_{i} \in \mathcal{O}_{\mathbb{F}_{n}}(a_{i},b_{i})\). Denote by \(u,v,z,w\) the coordinates for the Hirzebruch surface \(\mathbb{F}_{n}\), and suppose that \(\{f_{1}, \dots, f_{s}\}\) containes bases for \(\mathcal{O}_{\mathbb{F}_{n}}(1,0)\) and \(\mathcal{O}_{\mathbb{F}_{n}}(0,1)\). Then \(R(\mathbb{F}_{n},D)\) is generated in degree at most \[ \rho' = \text{LCM}_{1\leq i \leq s}(k_{i})\cdot \bigg( \sum_{1 \leq i\leq j \leq s} a_{i}b_{j}\bigg) \] with relations generated in degree at most \(2\rho '\).