Arquile Varieties – Varieties Consisting of Power Series in a Single Variable

Abstract

Spaces of power series solutions $y(\mathrm {t})$ in one variable $\mathrm {t}$ of systems of polynomial, algebraic, analytic or formal equations $f(\mathrm {t},\mathrm {y})=0$ can be viewed as ‘infinite-dimensional’ varieties over the ground field $\mathbf {k}$ as well as ‘finite-dimensional’ schemes over the power series ring $\mathbf {k}[[\mathrm {t}]]$ . We propose to call these solution spaces arquile varieties, as an enhancement of the concept of arc spaces. It will be proven that arquile varieties admit a natural stratification ${\mathcal Y}=\bigsqcup {\mathcal Y}_d$ , $d\in {\mathbb N}$ , such that each stratum ${\mathcal Y}_d$ is isomorphic to a Cartesian product ${\mathcal Z}_d\times \mathbb A^{\infty }_{\mathbf {k}}$ of a finite-dimensional, possibly singular variety ${\mathcal Z}_d$ over $\mathbf {k}$ with an affine space $\mathbb A^{\infty }_{\mathbf {k}}$ of infinite dimension. This shows that the singularities of the solution space of $f(\mathrm {t},\mathrm {y})=0$ are confined, up to the stratification, to the finite-dimensional part.

Keywords

• 13P10
• 13J05
• 14Q20
• 16W60
• 30H50
• 32A05
• 32A38