Computations and ML for surjective rational maps

Abstract

The present note studies \emph{surjective rational endomorphisms} $f:{\mathbb{P}}^2⇢{\mathbb{P}}^2$ with \emph{cubic} terms and the indeterminacy locus $I_f\ne \varnothing$. We develop an experimental approach, based on some Python programming and Machine Learning, towards the classification of such maps; a couple of new explicit $f$ is constructed in this way. We also prove (via pure projective geometry) that a general non-regular cubic endomorphism $f$ of ${\mathbb{P}}^2$ is surjective if and only if the set $I_f$ has cardinality at least $3$.

15 pages, 2 figures, a couple of Python codes, v2: minor corrections