Chow rings and gonality of general abelian varieties

Abstract

We study the (covering) gonality of abelian varieties and their orbits of zero-cycles for rational equivalence. We show that any orbit for rational equivalence of zero-cycles of degree $k$ has dimension at most $k-1$. Building on the work of Pirola, we show that very general abelian varieties of dimension $g$ have covering gonality $k\ge f(g)$ where $f(g)$ grows like $\log g$. This answers a question asked by Bastianelli, De Poi, Ein, Lazarsfeld and B. Ullery. We also obtain results on the Chow ring of very general abelian varieties, eg. if $g\ge 2k-1$, for any divisor $D\in {\mathrm{Pic}}^0(A)$, $D^k$ is not a torsion cycle.

Cited by in corpus (1)

• Covering gonalities of complete intersections in positive characteristic