TITLE: Examples of families of stable genus 2 curves
QUESTION [9 upvotes]: Given a smooth genus 2 curve $C$, it is canonically a two-fold cover of a $\mathbb{P}^1$, branched at six points. Allowing stable curves allows degenerations in two directions: the six points are allowed to collide (in certain ways), and $\mathbb{P}^1$ can "break" into a degenerate conic in $\mathbb{P}^2$ (here a union of two distinct lines). For instance, a smooth genus one curve with two of its points glued together is an example of the former situation, while two genus one curves joined to each other along a point is an example of the latter.
It is possible to construct explicit families (ideally, over high-dimensional bases and without appealing to stable reduction results) of stable genus 2 curves  that include degenerations of both kinds? (I would even be interested in degeneration of smooth curves to the second locus of unions of elliptic curves.) One difficulty here is that, if $C$ is a union of elliptic curves, then the map from $C$ to the quotient of $C$ by its "hyperelliptic" involution (multiplication by $-1$ on each elliptic curve) is not flat at the singular point.

REPLY [4 votes]: An explicit example of such a family with $1$-dimensional basis can be found in my paper  On surfaces of general type with $p_g=q=1, K^2=3$. The relevant statement is Proposition 6.1, that  I will restate here for the reader's convenience.

Proposition. There exists exactly one irreducible family of minimal surfaces $S$ of
  general type  with $p_g=q=1$, $K^2=3$ and a rational pencil $|G|$
  of curves of genus $2$, and this family is parametrized by the
  coarse moduli space of elliptic curves. Moreover:

the pencil $|G|$ is base point free, hence it defines a fibration $f \colon S \to \mathbb{P}^1$ whose general fibre is a smooth genus $2$ curve;
$|G|$ is the only genus $2$ pencil on $S$;
$|G|$ contains $13$ singular elements; six of these are genus $1$ curves
  with an ordinary double point, the other seven consist of two
  smooth elliptic curves intersecting transversally at a single
  point.


As far as I know this family was first constructed by Xiao Gang in his book Surfaces fibrées en courbes de genre deux (Lecture Notes in Mathematics 1137, Springer 1985), see in particular Chapter 3. The costruction follows from the classification of non-isotrivial genus $2$ fibrations $f \colon S \to C$ ($S$ smooth surface, $C$ smooth curve) whose associated Jacobian fibration has a fixed part $E \times C$. 
In Xiao's book one can also find many other examples, see in particular the table at page $52$. The family of surfaces in the Proposition corresponds to the case $d=4$ in that table. These surfaces are explicitly mentioned in the Corollaire $4$, page $51$, where they are called fibrations $f(E, 4)$.