Nef cone volumes and discriminants of abelian surfaces

The nef cone volume appeared first in work of Peyre in a number-theoretic context on Fano varieties, and was then studied by Derenthal and co-authors in a series of papers on del Pezzo surfaces. The idea was subsequently extended to also measure the Zariski chambers of del Pezzo surfaces. We start in this paper to explore the possibility to use this attractive concept to effectively measure the size of the nef cone on algebraic surfaces in general. This provides an interesting way of measuring in how big a space an ample line bundle can be moved without destroying its positivity. We give here complete results for simple abelian surfaces that admit a principal polarization and for products of elliptic curves.