I'm stumped by Exercise 2.3 in Harris-Morrison, which says:
"Show that there does not exist a universal family of curves of genus 2 over any open subset $U \subset M_2$. In general, if $H_g \subset M_g$ is the locus of hyperelliptic curves, for which $g$ does there exist a universal family over some open subset $U \subset H_g$? Answer: for $g$ odd."
It's not 100% clear what is meant by "universal family"; possibly it just means some family each of whose (closed) fiber is ``the right curve''.
Question: what precisely is the question, and then what is the answer?
In any case the "exercise" seems to suggest a fundamental difference between hyperelliptic curves with odd vs even genus. I'd be happy to see an explanation of what this difference is.