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The tropical 1-fold Abel-Prym map
January 17 at 14:30 – 15:30 CET
TGiF-Seminar: Tropical geometry in Frankfurt (First meeting Winter Semester 2024/25)
Giusi Capobianco (Università di Roma Tor Vergata)
Abstract: The algebraic Abel-Prym map relates the geometry of a double cover of algebraic curves with their corresponding Prym varieties. Birkenhake and Lange proved that the map has degree 2 if and only if the cover curve is hyperelliptic.
In the talk I will present joint work with Yoav Len, in which we investigate the 1-fold Abel-Prym map in the tropical setting and prove similar results. I will describe a new combinatorial construction of hyperelliptic double covers of metric graphs and prove that the tropical Abel-Prym map is a harmonic morphism of degree 2. Furthermore, we will see that the Jacobian of the image of this map is isomorphic, as pptav, to the Prym variety of the cover. When the double cover is not hyperelliptic however, contrary to the algebraic result, the tropical Abel-Prym map is almost never injective. I will provide counterexamples and discuss its image.