Given a class of curves, is there one uniform way of writing these down? For elliptic curves, the answer is yes: over the complex numbers, any elliptic curve can be uniquely written in the form
y^2 = x^3 + a x + b.
We consider the abstract meaning of our question, and introduce the classical notion of a universal families of curves. These are very well-behaved families that give an optimal way to write down curves in a given class; the problem is that they usually do not exist.
We explain how to get around this problem by defining a more general analogue of the pleasant family of elliptic curves above; more precisely, these are families induce morphisms to the moduli space that are bijective over the algebraic closure. We then construct some other concrete families of this kind for the special case of plane quartic curves, and give some applications to the study of curves over finite fields.
This is joint work with Reynald Lercier, Christophe Ritzenthaler, and Florent Rovetta.