To any graph and smooth algebraic curve $C$ one may associate a "hypercurve"arrangement and one can study the rational homotopy theory of the complement$X$. In the rational case ($C=\mathbb{C}$), there is considerable literature onthe rational homotopy theory of $X$, and the trigonometric case ($C =\mathbb{C}^\times$) is similar in flavor. The case of when $C$ is a smoothprojective curve of positive genus is more complicated due to the lack offormality of the complement. When the graph is chordal, we use quadratic-linearduality to compute the Malcev Lie algebra and the minimal model of $X$, and weprove that $X$ is rationally $K(\pi,1)$.
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机译:对于任何图和平滑的代数曲线$ C $,都可以关联“超曲线”排列,并且可以研究补数$ X $的有理同伦理论。在有理情况下($ C = \ mathbb {C} $),关于$ X $的有理同伦理论的文献很多,而三角情况($ C = \ mathbb {C} ^ \ times $)在味道。 $ C $是正属的平滑投影曲线时,由于缺乏补语的形式性,因此情况更为复杂。当图是弦图时,我们使用二次线性对偶来计算Malcev Lie代数和$ X $的最小模型,并证明$ X $合理地是$ K(\ pi,1)$。
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