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arXiv · 1909.01267

Cox rings of K3 surfaces of Picard number three

Michela Artebani·Claudia Correa·Antonio Laface
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Abstract

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.

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Michela Artebani, Claudia Correa, Antonio Laface. 2019-09-03. Cox rings of K3 surfaces of Picard number three. https://arxiv.org/abs/1909.01267

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