Fractional Dehn twists and modular invariants

Abstract

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.