Abstract
In this short note, I point out that results of Ballico and Kool–Shende–Thomas together imply that on K3, Enriques, and Abelian surfaces, if L is a very ample and \((2p_a(L)-2g-1)\)-spanned line bundle, then the equigeneric Severi variety \(V_{g}(L)\) of all curves in |L| having genus g is non-empty, irreducible, of the expected dimension, and its general member is a \((p_a(L)-g)\)-nodal curve.
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Dedieu, T. Comment on: On the irreducibility of the Severi variety of nodal curves in a smooth surface, by E. Ballico. Arch. Math. 114, 171–174 (2020). https://doi.org/10.1007/s00013-019-01392-9
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DOI: https://doi.org/10.1007/s00013-019-01392-9