Computations and ML for surjective rational maps

TL;DR

The paper investigates surjective rational endomorphisms of $\mathbb{P}^2$ with cubic terms and nonempty indeterminacy loci, linking birational geometry to dynamical properties. It combines a rigorous geometric analysis—via del Pezzo surfaces and linear projections—with an experimental framework that samples maps over finite fields and uses data-driven predictors to guide the search for explicit examples. A key theoretical result is that for general non-regular cubic endomorphisms, surjectivity is equivalent to $|I_f| \ge 3$ in the generic regime, while for $\delta \ge 3$ the surjectivity holds for general $f$ through a regular projection from the anticanonically embedded del Pezzo surface $X_\delta$; for $\delta \le 2$, an unruly pencil obstructs surjectivity. The work provides concrete surjective examples, demonstrates a practical computational approach to classify such maps, and sketches a framework that blends classical algebraic geometry with data-driven search techniques to advance understanding of rational maps in projective planes.

Abstract

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 $3$.

Figures (2)

• Figure 1: Two curves $C$ and $Z$
• Figure 2: Projection onto $\psi(E_i)$

Theorems & Definitions (13)

• Remark 1.2
• Remark 1.4
• Proposition 2.2
• proof
• Corollary 2.3
• Theorem 2.6
• proof
• Lemma 2.7
• proof
• Proposition 3.2