Considering a smooth genus two curve $C_2$, let $J(C_2)$ be its Jacobian surface, and take $p \in J(C_2)$ an $m$-torsion point. Let $A = J(C_2)/Z_m$, where $Z_m$ acts by $x \mapsto x+p$. The image of $C_2$ under the composition $C_2 \xrightarrow{\mathrm{Abel-Jacobi}} J(C_2) \xrightarrow{/Z_m} A$ is a genus two curve $C$ with $2m-2$ self-intersections. The linear system $L = \mathscr{O}(C)$ establishes a mapping $A \to \mathbf{P}^{2m-1}$. When $m=2$, for instance, the image cannot be smooth since otherwise it would be an abelian surface in $\mathbf{P}^3$, which is forbidden by e. g. Lefschetz hyperplane theorem. Nevertheless, the generic plane section of this image is a smooth genus four curve. The singular sections are the same as tangent planes, and the number of singularities of a curve cut out by a plane induces a stratification on the dual projective space $\check{\mathbf{P}}^3 = \mathbf{P}(H^0(L))$.
My question is: what does this stratification look like? It contains a two-parametric family of planes cutting out genus three curves with one singularity -- these are the planes passing through the singular points of the image of $A$; but these are not the planes in the image of the Gauss map from $A$ to $\mathbf{P}(H^0(L))$. Is it true that any plane which is tangent to $A$ in a smooth point is necessarily a bitangent (i. e. the homology class of the section cannot be represented by a genus three curve with one self-intersection other than the singular points of the image of $A$)? If not, is the dimension of the locus of bitangents zero or one? If latter, can one describe this curve geometrically? Or maybe I am completely misled and the map $A \to \mathbf{P}^3$ is not even generically injective, i. e. factorizes through the involution $x\mapsto -x$ or whatever?
My objective is to describe the analogous stratification on $\mathbf{P}(H^0(L))$ for $m > 2$. Thus another question is: is the aforementioned mapping $A \to \mathbf{P}^{2m-1}$ an embedding for some value $m > 2$? The problem appears quite classical, so I would appreciate any reference on this stratification in this case.
