For a family of Jacobians of smooth pointed curves there is a notion of tautological algebra. There is an action of $\mathfrak{sl}_2$ on this algebra. We define and study a lifting of the Polishchuk operator, corresponding to $f\in \mathfrak{sl}_2$, on an algebra consisting of punctured Riemann surfaces. As an application we prove that a collection of tautological relations on moduli of curves, discovered by Faber and Zagier, come from a class of relations on the universal Jacobian.

中文翻译：

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通过穿孔表面的Polishchuk微分算子的作用

对于平滑尖曲线的雅可比族，存在重言式代数的概念。在这个代数上有 $\mathfrak{sl}_2$ 的作用。我们定义并研究了Polishchuk 算子的提升，对应于$f\in\mathfrak{sl}_2$，在由穿孔黎曼曲面组成的代数上。作为一个应用，我们证明了 Faber 和 Zagier 发现的关于曲线模的重言式关系的集合，来自于通用雅可比矩阵上的一类关系。