Tropical Donagi theorem

TL;DR

This paper extends the tropical $n$-gonal construction to $n=4$, proving a tropical Donagi's theorem that reveals triality preserving Prym varieties and discusses the construction's limitations under edge contractions.

Contribution

It establishes a tropical analogue of Donagi's theorem for $n=4$, confirming previous conjectures and analyzing the construction's behavior and limitations.

Findings

Proves tropical Donagi's theorem for $n=4$

Shows the triality preserves Prym varieties

Demonstrates poor behavior under edge contractions

Abstract

The tropical $n$-gonal construction was introduced in recent work by the first author and D.~Zakharov and structural results for $n = 2,3$ were established. In this article we explore the construction for $n = 4$ and prove a tropical analogue of Donagi's theorem which states that the tetragonal construction is a triality which preserves Prym varieties. This confirms the speculations in previous work and establishes new results on the non-injectivity of the tropical Prym-Torelli morphism. Finally, we demonstrate that the tropical $n$-gonal construction is poorly behaved under edge contractions, thus preventing any immediate moduli-theoretic perspective.