Constructions for rational multiple planes

TL;DR

This paper classifies simpler rational multiple planes with branch curves under certain conditions, showing they form infinitely many non-Cremona equivalent families and providing explicit examples for degrees four and above.

Contribution

It introduces a classification of simpler triple planes and constructs infinitely many non-Cremona equivalent families of multiple planes of degree at least four.

Findings

Classified simpler triple planes up to Cremona equivalence.

Constructed infinitely many non-Cremona equivalent families.

Provided explicit examples for degrees ≥ 4.

Abstract

A finite, normal cover $f: X\longrightarrow \bbP^2$ of degree $m\geq 3$ (the case $m=2$ is well known and we do not consider it in this paper) is called \emph{simple}, if there is a pencil $\mathcal P$ of rational curves of $\bbP^2$ such that the pull back via $f$ of $\mathcal P$ is a pencil of rational curves on $X$. Up to Cremona equivalence $\mathcal P$ can be assumed to be the pencil of lines through a fixed point $p\in \bbP^2$. If $\frakB$ is the branch curve of such a multiple plane, the general line through $p$ has to intersect $\frakB$ in $2m-2$ branch points (counted with multiplicities). If $p$ is not one of these branch points, then the multiple plane is said to be \ \emph{simpler}. \ In that case the branch curve will have a point of multiplicity $\deg(\frakB)-2m+2$ at $p$. In this paper we classify, under suitable generality conditions for the branch curve, { simpler } triple planes up to Cremona equivalence (they belong to infinitely many non--Cremona equivalent families) and we give examples of infinitely many non--Cremona equivalent families of {simpler } multiple planes of degree $m\geq 4$.