  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.
