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.



Neron model: can number of components decrease after based change?
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.


