2017/08/03 by Paweł Borówka, Angela Ortega · 1 citation
Mathematics · #math.AG #msc:14H40 #msc:14H30
published as Math Z. vol. 292, Issue 1-2 (2019), 193-209 · 17 pages
arxiv created 2017/08/03 · arxiv updated 2019/11/13
We investigate the number and the geometry of smooth hyperelliptic curves on a general complex abelian surface. We show that the only possibilities of genera of such curves are 2,3,4 and 5. We focus on the genus 5 case. We prove that up to translation, there is a unique hyperelliptic curve in the linear system of a general (1,4) polarised abelian surface. Moreover, the curve is invariant with respect to a subgroup of translations isomorphic to the Klein group. We give the decomposition of the Jacobian of such a curve into abelian subvarieties displaying Jacobians of quotient curves and Prym varieties. Motivated by the construction, we prove the statement: every étale Klein covering of a hyperelliptic curve is a hyperelliptic curve, provided that the group of 2-torsion points defining the covering is non-isotropic with respect to the Weil pairing and every element of this group can be written as a difference of two Weierstrass points.