Computations and ML for surjective rational maps
This work addresses a specific problem in algebraic geometry for researchers, but it is incremental as it builds on existing theory with experimental methods.
The paper tackles the classification of surjective rational endomorphisms with cubic terms in projective space, constructing new explicit maps and proving a condition for surjectivity based on the indeterminacy locus cardinality.
The present note studies \emph{surjective rational endomorphisms} $f: \mathbb{P}^2 \dashrightarrow \mathbb{P}^2$ with \emph{cubic} terms and the indeterminacy locus $I_f \ne \emptyset$. 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 $3$.