arXiv subjects
Publications and source records attributed to Giusi Capobianco.
We give a new intrinsic construction of the continuous tropical Prym variety, originally due to R\"ohrle and Zakahrov. In particular, given a harmonic double cover of graphs we explicitly construct the tropical Prym as integral torus. Moreover, we show that the tropical Prym--Torelli map from the moduli space of harmonic double covers to the moduli space of principally polarised tropical abelian varieties is continuous.
We study the conjecture stated by Jensen and Len on a tropical version on Martens' theorem via the Brill--Noether rank of a tropical curve. We recall Coppens' counterexample of Martens-special chain of cycles, and we generalize the construction defining another class of graphs, Martens-special trees of cycles, for which the conjecture does not hold in a similar setting. These are not the only counterexamples. However, we prove that the conjecture holds for all metric graphs with a stricter assumption on the degree in the Brill--Noether rank.
We prove that the tropical Abel--Prym map $\Psi\colon \widetilde\Gamma\to Prym(\widetilde\Gamma/\Gamma)$ associated with a free double cover $\pi\colon \widetilde\Gamma\to \Gamma$ of hyperelliptic metric graphs is harmonic of degree $2$ in accordance with the already established algebraic result. We then prove a partial converse. Contrary to the analogous algebraic result, when the source graph of the double cover is not hyperelliptic, the Abel--Prym map is often not injective. When the source graph is hyperelliptic, we show that the Abel--Prym graph $\Psi(\widetilde\Gamma)$ is a hyperelliptic metric graph of genus $g_{\Gamma}-1$ whose Jacobian is isomorphic, as pptav, to the Prym variety of the cover. En route, we count the number of distinct free double covers by hyperelliptic metric graphs.