纤维化与模空间的研究都对代数簇的分类有着重要作用，而且二者紧密联系在一起。模不变量是联系曲面纤维化和曲线模空间的一个数值不变量，在数学的各个分支都有不同的体现。本项目主要研究以下几个问题：1.在已有工作的基础上，估计模不变量的精确下界，并分类达到下界的纤维化；2. 完善关于模空间中与边界除子有特定相交关系的曲线的存在性；3. 对于i-型可分曲线，确定Dehn twist 根的次数极值，以及达到极值的根的分类。

Fibrations and moduli spaces are important in the classification of algebraic varieties, and they are closely linked with each other. Modular invariants are numerical invariants connecting fibrations of algebraic surfaces with moduli spaces of curves, and they have different presentations in various branches of mathematics. This project focuses on the following three questions: 1. based on the work before, we want to obtain the best lower bounds of moduli invariants, and classify the fibrations with the bounds. 2. complete the previous work on the existence of curves in moduli spaces with prescribed intersection with boudary divisors. 3. for separating curves of type i, we hope to give the optimum bounds of the degrees of the roots of Dehn twists, and classify the roots reaching the bounds.