TITLE: A curve with bad reduction for which the jacobian has good reduction
QUESTION [13 upvotes]: Let $K$ be a number field. If $X$ is a curve over $K$ with good reduction at a place $v$ of $K$, then the Jacobian of $X$ also has good reduction at $v$. This follows from the functoriality of the Jacobian.
The converse is not true, but I don't know of any examples.
Can one provide an example for all number fields $K$? 
If not, I would also be pleased with just a counterexample for some place $v$ of some number field $K$

REPLY [13 votes]: Let $E,E'$ be elliptic curves over the residue field.  Then $E \times E'$
has good reduction but is not a Jacobian.  However, any principally
polarized abelian surface that reduces to $E \times E'$ but is not
a product of elliptic curves is a Jacobian of some genus-$2$ curve $C$
that cannot have good reduction.
Explicitly (when the residue characteristic is odd), $C$ can have the form $y^2 = P(x)$ where $P$ is a sextic
with roots $x_1,x_2,x_3$, $1/x'_1,1/x'_2,1/x'_3$ 
such that each $x_i,x'_i$ reduces to zero and each
$x_i-x_j$ and $x'_i-x'_j$ ($i\neq j$) has valuation $1$.
If I remember right, the Jacobian reduces to $E \times E'$
where $E: Y^2 = (X-\bar x_1) (X-\bar x_2) (X-\bar x_3)$
and $E': Y^2 = (X-\bar x'_1) (X-\bar x'_2) (X-\bar x'_3)$.
As I recall I learned this construction from Joe harris.