On the sharp lower bounds of modular invariants and fractional Dehn twist coefficients

Modular invariants of families of curves are Arakelov invariants in arithmetic algebraic geometry. All the known uniform lower bounds of these invariants are not sharp. In this paper, we aim to give explicit lower bounds of modular invariants of families of curves, which is sharp for genus 2. 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 for genus 2 and 3 firstly. We also obtain a rigidity property for families with minimal modular invariants, and other applications.