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Birational complexity of log Calabi-Yau 3-folds

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arxiv 2405.18516 v1 pith:L7A4R6YP submitted 2024-05-28 math.AG

classification math.AG
keywords birationalcalabi-yaucomplexitycrepantfoldsmathbbproveadmits
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abstract

We study the birational complexity of log Calabi-Yau $3$-folds. For such a pair $(X,B)$ of index one and coregularity zero, we show that $c_{\rm bir}(X,B)\in \{0,2,3\}$. Further, we prove that $(X,B)$ has a log Calabi-Yau crepant birational model that admits a crepant contraction to $(\mathbb{P}^{3-c},H_0+\dots+H_{3-c})$, where $c=c_{\rm bir}(X,B)$. To prove this, we give a geometric characterization of standard $\mathbb{P}^1$-links.

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Cited by 2 Pith papers

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  1. Complexity one varieties are cluster type

    math.AG 2025-04 conditional novelty 7.0 of 10

    A log Calabi-Yau pair of index one and complexity at most one is a cluster type variety, and varieties of absolute complexity one admit a finite cluster type cover of degree at most two.

  2. Calabi-Yau pairs of complexity two

    math.AG 2024-12 conditional novelty 7.0 of 10

    A Calabi-Yau pair of complexity two is cluster type exactly when its standard model over a toric base is nodal, component-compatible, and volume-large; this classifies all rank-one Gorenstein del Pezzo surfaces.

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