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Suppose I have Neron model over some discrete valuation ring.

Is there a result such that the number of components of the fiber over the closed point cannot decrease after some based change?

In particular, in the case when it is the Neron model over discrete valuation ring $R$ of Jacobian of the model over $R$ of Hyperelliptic curve which have stable reduction over the closed point.

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    $\begingroup$ Minor nitpick: if you base-change the Neron model, then (obviously) the number of components can not decrease. What you are really asking is whether, for $X$ an abelian variety over $K$ (the function field of $R$, say) and $L/K$ finite field extension, the number of connected components of the Neron model of $X_L$ can be smaller than the number of connected components of the Neron model of $X$. As Kestutis Cesnavicius points out, this can happen unless you have semi-abelian reduction (as in the last part of your question). $\endgroup$
    – Ariyan Javanpeykar
    Commented Jul 16, 2014 at 13:19

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Yes, there is such a result for Neron models of abelian varieties with semiabelian (aka semistable) reduction: "the number of components of the special fiber cannot decrease after base change". This is a special case of Prop. 3 in section 7.4 of Bosch, Lutkebohmert, Raynaud "Neron models".

Beyond the semiabelian case though, a similar claim is false: consider, for instance, the Neron model of an elliptic curve with potential good reduction.

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