We give an explicit formula of the normalized Mumford form which expresses the second tautological line bundle by the Hodge line bundle defined on the moduli space of algebraic curves of any genus. This formula is represented as an infinite product which is a higher genus version of the Ramanujan delta function under the trivialization by normalized abelian differentials and Eichler integrals of their products. Furthermore, this formula gives a universal expression of the normalized Mumford form as a computable power series with integral coefficients by the moduli parameters of algebraic curves. Therefore, one can describe the behavior of this form and hence of the Polyakov string measure around the Deligne-Mumford boundary.

中文翻译：

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归一化 Mumford 形式的显式公式

我们给出了一个明确的归一化芒福德形式的公式，它通过定义在任何属的代数曲线的模空间上的霍奇线丛来表达第二重言式线丛。该公式表示为无限乘积，它是 Ramanujan delta 函数在归一化阿贝尔微分和其乘积的 Eichler 积分的平凡化下的更高属版本。此外，该公式给出了归一化 Mumford 形式的通用表达式，即具有由代数曲线的模参数的积分系数的可计算幂级数。因此，我们可以描述这种形式的行为，从而描述围绕德利涅-芒福德边界的 Polyakov 弦测度的行为。