Abel maps for nodal curves via tropical geometry

Abstract: Let $\pi\colon \mathcal{C}\rightarrow B$ be a regular smoothing of a nodal curve with smooth components and a section $\sigma$ of $\pi$ through its smooth locus. Let $\mu$ and $\mathcal{L}$ be a polarization and an invertible sheaf of degree $k$ on $\mathcal{C}/B$. The Abel map $\alpha^{d}_{\mathcal{L}}$ is the rational map $\alpha^{d}_{\mathcal{L}}\colon \mathcal{C}^d \dashrightarrow \overline{\mathcal{J}}_{\mu}^{\sigma}$ taking a tuple of points $(Q_1,\dots,Q_d)$ on a fiber $C_b$ of $\pi$ to the sheaf $\mathcal{O}_{C_b}(Q_1+\dots+Q_d-d\sigma(b))\otimes \mathcal{L}|_{C_b}$. Here $\overline{\mathcal{J}}_{\mu}^{\sigma}$ denotes Esteves compactified Jacobian. An interesting question is to find an explicit resolution of the map $\alpha^{d}_{\mathcal{L}}$. We translate this problem into an explicit combinatorial problem by means of tropical and toric geometry. The solution of the combinatorial problem gives rise to an explicit resolution of the Abel map. We are able to use this technique to construct all the degree-$1$ Abel maps and give a resolution of the degree-$2$ Abel-Jacobi map.

algebraic geometrycombinatorics

Audience: researchers in the topic

(LAGARTOS) Latin American Real and Tropical Geometry Seminar

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