Let \(X^{1,n}_r\) be the blow-up of \(\mathbb {P}^1\times \mathbb {P}^n\) in r general points. We describe the Mori cone of \(X^{1,n}_r\) for \(r\le n+2\) and for \(r = n+3\) when \(n\le 4\). Furthermore, we prove that \(X^{1,n}_{n+1}\) is log Fano and give an explicit presentation for its Cox ring.

中文翻译：

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多射影空间爆炸的考克斯环

设\(X^{1,n}_r\)为r 个一般点中\(\mathbb {P}^1\times \mathbb {P}^n\)的放大。我们描述\(r\le n+2\)和\(r = n+3\)当\(n\le 4\)时的\(X^{1,n}_r\)的森锥。此外，我们证明了\(X^{1,n}_{n+1}\)是 log Fano 并给出了其 Cox 环的显式表示。