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!

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!

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 $p$? (those form has singular point at infinity when genus is larger than 1, so let's assume that we took those normalization before considering those Jacobian)

Thank you in advance!

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!