For general, let X be an abelian variety of dimension g. We say that X has 'purely additive reduction' at prime p if the dimension of the unipotent radical of the special fiber of the Neron Model of X is equal to g.

I have two questions.

• How do we practically check if a Jacobian of hyperelliptic curve has purely additive reduction at some place v. for example, Prof. Liu Qing pointed out (on the following post) Jacobian of $y^2 = x^{2g+1} + f$ $(f$ is uniformizing element of some prime $p$) has purely additive reduction at $p$

Can we always find a curve which doesn't have semi-stable reduction

But I don't know how to check it.

• Let the Jacobian of $y^2 = x^{2g+1} + a_{2g}x^{2g} + ... + a_{0} $(defined over some number field $K$) has good reduction at $p$. And let $d$ be a uniformizing element of prime $p$. Then the Jacobian of $dy^2 = x^{2g+1} + a_{2g}x^{2g} + ... + a_{0} $ has purely additive reduction at $p$?. In other words, the Jacobian of $y^2 = x^{2g+1} + d^{1}a_{2g}x^{2g} + ... + d^{2g+1}a_{0} $ has purely additive reduction at $p$?

Thank you in advance!