Rationality problem of three-dimensional monomial group actions

Abstract

Let $K$ be a field of characteristic not two and $K(x,y,z)$ the rational function field over $K$ with three variables $x,y,z$. Let $G$ be a finite group of acting on $K(x,y,z)$ by monomial $K$-automorphisms. We consider the rationality problem of the fixed field $K(x,y,z{)}^G$ under the action of $G$, namely whether $K(x,y,z{)}^G$ is rational (that is, purely transcendental) over $K$ or not. We may assume that $G$ is a subgroup of $\mathrm{GL}(3,\mathbb{Z})andtheproblemisdetermineduptoconjugacyin$\mathrm{GL}(3,\mathbb{Z})$.Thereare73conjugacyclassesof$G$in$\mathrm{GL}(3,\mathbb{Z})$.ByresultsofEndo-Miyata,Voskresenski\breve{ı},Lenstra,Saltman,Hajja,KangandYamasaki,8conjugacyclassesof2-groupsin$\mathrm{GL}(3,\mathbb{Z})$havenegativeanswerstotheproblemundercertainmonomialactionsoversomebasefield$K$,andthenecessaryandsufficientconditionfortherationalityof$K(x,y,z)^G$over$K$isgiven.Inthispaper,weshowthatthefixedfield$K(x,y,z)^G$undermonomialactionof$G$isrationalover$K$exceptforpossiblynegative8casesof2-groupsandunknownonecaseofthealternatinggroupofdegreefour.Moreoverwegiveexplicittranscendentalbasesofthefixedfieldsover$K$.Forunknowncase,weobtainanaffirmativesolutiontotheproblemundersomeconditions.Inparticular,weshowthatif$K$isquadraticallyclosedfieldthen$K(x,y,z)^G$isrationalover$K$. We also give an application of the result to 4-dimensional linear Noether's problem.

54 pages

References in corpus (2)

• The rationality problem for fields of invariants under linear algebraic groups (with special regards to the Brauer group)
• The rationality problem for finite subgroups of GL_4(Q)