Purely additive reduction of Jacobian of Hyperelliptic curve

Question

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!

Answer 1

It is all about the $g$-cuspidal singularity $y^2 = x^{2g+1}$ of the special fibre, and how this singularity affects the Picard group in terms of the Picard group of the normalization (which is an abelian variety in general, and is trivial in the case you consider).

Consider more generally a (connected, reduced, but) singular curve $C$, and its normalization $\pi : \widetilde{C} \to C$. The Picard groups of $C$ and $\widetilde{C}$ are the $H^1$s of the sheaves $\mathcal{O}_C^{\times}$ and $\pi_* \mathcal{O}_{\widetilde{C}}^{\times}$ on $C$, respectively, and the relation between the two is governed by the sheaf cokernel of the inclusion $\mathcal{O}_C^{\times} \to \pi_* \mathcal{O}_{\widetilde{C}}^{\times}$. The latter is a skyscraper sheaf supported at the singular points of $C$; it is therefore determined completely by the singularity types in a purely local manner. In your example of a $g$-cuspidal singularity $p \in C$, denoting $t$ a local parameter at the unique point on $\widetilde{C}$ above $p$, the completion of the stalk of $\pi_*\mathcal{O}_{\widetilde{C}}^{\times}$ at $p$ consists of the power series $a_0 + a_1t + a_2t^2 + \cdots$ with $a_0 \neq 0$, while the completion of the stalk of $\mathcal{O}_{C}^{\times}$ at $p$ is the subgroup with $a_1 = \cdots = a_g = 0$. Therefore the cokernel has stalk $k^g = \mathbb{G}_a^g(k)$ at such a singularity $p$, $k$ being the ground (or residue) field. Do this at every singular point to find the $H^0$ of the sheaf cokernel $\mathcal{F}$ (the $H^1$ vanishes), and apply cohomology to $0 \to \mathcal{O}_C^{\times} \to \pi_* \mathcal{O}_{\widetilde{C}}^{\times} \to \mathcal{F} \to 0$ to get the description of $\mathrm{Pic}(C) = H^1(\mathcal{O}_C^{\times})$.

This point is explained in good detail in section [5. B: Limits of line bundles] of the book "Moduli of curves" by Harris and Morrison. Your particular question is given as Exercise 5.14 (2). Also see 5.11 (2) in particular.