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Let $J$ be the Jacobian of a smooth projective connected curve of genus $g>1$ over the field $\overline{\mathbf{Q}}$ of algebraic numbers.

Does $J$ have everywhere good reduction?

I know that there are abelian varieties of any dimension which do not have good reduction everywhere. Just take a suitable elliptic curve $E$ and consider the product $E^g$.

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It needn't have everywhere good reduction, no. For instance if $C_{/\mathbb{Q}}$ is a curve which has semistable bad reduction at a prime $p$, then this reduction stays semistable bad over any algebraic extension.

For easy examples, take any modular curve $X_0(p)$ for any prime number $p \geq 23$. Such curves have genus at least $2$ and semistable bad reduction at $p$.

More generally, it is easy to construct for each $g \geq 2$ and prime number $p$ a curve $C_{/\mathbb{Q}}$ of genus $g$ with semistable bad reduction at $p$. Examples of curves over $\mathbb{Q}_p$ (and also defined over some number field $K$ and thus also over $\overline{\mathbb{Q}}$) with these properties (and others...) are constructed in this paper of mine.