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The modular invariants of a family of semistable curves are the degrees of the corresponding divisors on the image of the moduli map. The singularity indices were introduced by G. Xiao to classify singular fibers of hyperelliptic fibrations and to compute global invariants locally. In the semistable case, we show that the modular invariants corresponding with the boundary classes are just the singularity indices. As an application, we show that the formula of Xiao for relative Chern numbers is the same as that of Cornalba-Harris in the semistable case.

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In this paper, we aim to give explicit lower bounds of modular invariants of families of curves of genus 2 and 3. According to the relation between fractional Dehn twists and modular invariants, we give the sharp lower bounds of fractional Dehn twist coefficients and classify pseudo-periodic maps with minimal coefficients firstly. Then we give explicit lower bounds of modular invariants, which are sharp for genus 2. We also obtain equivalent conditions for families of curves with these bounds. As an application, we give a better uniform lower bounds for the effective Bogomolov conjecture for genus 2 and 3.

We construct curves in moduli spaces of curves with prescribed intersection with boundary divisors. As applications, we obtain families of curves with maximal slope as well as extremal test curves for the weakly positive cone of the moduli space.Let ᏹ 0,n be the moduli space of stable unordered n-pointed rational curves. Let B k be the boundary divisor of ᏹ 0,n whose general point parametrizes the union of a k-pointed ‫ސ‬ 1 and anOne can regard Ᏼ g as the Hurwitz space parametrizing genus g admissible double covers of rational curves. Such a cover uniquely corresponds to a stable (2g + 2)-pointed rational curve by marking the branch points of the cover. Thus Ᏼ g can be further identified as ᏹ 0,2g+2 . The natural isomorphism Ᏼ g ∼ = ᏹ 0,2g+2 induces the identifications i = B 2i+2 and i = B 2i+1 . Also denote by Ꮾ = {B 2 , B 3 , . . . , B [n/2] } the set of boundary divisors of ᏹ 0,n . Hence the existence of curves in Ᏼ g in Problem 1.1 is the same as that in ᏹ 0,n for n = 2g + 2. For the sake of completeness, we consider the existence of curves in ᏹ 0,n . Precisely, we consider the following problem. Problem 1.2. For any nonempty subset Ꮾ ⊆ Ꮾ, does there exist a curve C in ᏹ 0,n such that the boundary divisors of ᏹ 0,n that intersect C are those in Ꮾ , i.e., such that Ꮾ C = Ꮾ ?The purpose of this paper is to answer these problems in a number of cases. We have the following uniform solution for small n.Theorem 1.3. Assume n ≤ 17. For any nonempty subset Ꮾ of Ꮾ, there exists a curve C in ᏹ 0,n such that Ꮾ C = Ꮾ .Unfortunately, our method is invalid for n = 18 (see Remark 6.1).Corollary 1.4. For 2 ≤ g ≤ 7 and any nonempty Ꮾ ⊂ Ꮾ, there exists a curve C in Ᏼ g such that Ꮾ C = Ꮾ . Corollary 1.5. For 2 ≤ g ≤ 7 and any nonempty Ꮾ ⊂ = { 0 , 1 , . . . , [g/2] }, there exists a curve C in ᏹ g with Ꮾ C = Ꮾ .