Voisin, Claire
Chow ring and gonality of general abelian varieties
Annales Henri Lebesgue, Volume 1 (2018), p. 313-332

KeywordsAbelian varieties, covering gonality, zero-cycles, Chow ring

### 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 at least $f\left(g\right)$, where $f\left(g\right)$ grows like $\mathrm{log}\phantom{\rule{0.166667em}{0ex}}g$. 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 $A$ of dimension $g$, e.g., if $g\ge 2k-1$, the set of divisors $D\in {\mathrm{Pic}}^{0}\left(A\right)$ such that ${D}^{k}=0$ in ${\mathrm{CH}}^{k}\left(A\right)$ is at most countable.

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