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Cox rings of K3 surfaces of Picard number three
Journal of Algebra ( IF 0.8 ) Pub Date : 2021-01-01 , DOI: 10.1016/j.jalgebra.2020.08.016
Michela Artebani , Claudia Correa Deisler , Antonio Laface

Let $X$ be a projective K3 surface over $\mathbb C$. We prove that its Cox ring $R(X)$ has a generating set whose degrees are either classes of smooth rational curves, sums of at most three elements of the Hilbert basis of the nef cone, or of the form $2(f+f')$, where $f,f'$ are classes of elliptic fibrations with $f\cdot f'=2$. This result and techniques using Koszul's type exact sequences allow to determine a generating set for the Cox ring of all Mori dream K3 surfaces of Picard number three which is minimal in most cases. A presentation for the Cox ring is given in some special cases with few generators.

中文翻译:

皮卡德三号 K3 表面的考克斯环

令 $X$ 是 $\mathbb C$ 上的一个射影 K3 曲面。我们证明它的 Cox 环 $R(X)$ 有一个生成集,其度数是平滑有理曲线的类,nef 锥的希尔伯特基的最多三个元素的总和,或者形式为 $2(f+f ')$,其中 $f,f'$ 是 $f\cdot f'=2$ 的椭圆纤维化类别。该结果和使用 Koszul 类型精确序列的技术允许确定 Picard 3 号的所有 Mori dream K3 表面的 Cox 环的生成集,这在大多数情况下是最小的。在一些生成器很少的特殊情况下给出了 Cox 环的演示。
更新日期:2021-01-01
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