Suppose I have a family of trigonal curves $C\to D$ over a closed disk $D$ where the central fiber $C_0$ is hyperelliptic (this is of course possible since the hyperelliptic locus is in the closure of the trigonal one inside the moduli space).
Without knowing that $C_0$ is hyperelliptic, can one guess this just by considering the surface/family $C$? That is, are there numerical features of $C$ that would imply that the limit is hyperelliptic? Even the opposite question should be interesting: are there features of $C$ that would imply that it is impossible that $C_0$ is hyperelliptic?
I am aware that the question is VERY open to different interpretations so please feel free to post remarks or possible improvements (and edits) of the question.
