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 has dimension at most . Building on the work of Pirola, we show that very general abelian varieties of dimension have (covering) gonality at least , where grows like . 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 of dimension , e.g., if , the set of divisors such that in is at most countable.
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[MZ17] On the group of zero-cycles of holomorphic symplectic varieties (2017) (https://arxiv.org/abs/1711.10045)
[Voi92] Sur les zéro-cycles de certaines hypersurfaces munies d’un automorphisme, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 19 (1992) no. 4, pp. 473-492 | Zbl 0786.14006
[Voi15a] Rational equivalence of 0-cycles on K3 surfaces and conjectures of Huybrechts and O’Grady, Recent advances in algebraic geometry (London Mathematical Society Lecture Note Series), Volume 417, Cambridge University Press, 2015, pp. 422-436 | Article | MR 3380459 | Zbl 1326.14089
[Voi16] Rational equivalence of 0-cycles on K3 surfaces and conjectures of Huybrechts and O’Grady, K3 surfaces and their moduli (Progress in Mathematics), Birkhäuser/Springer, 2016, pp. 365-399 | Zbl 1326.14089