Arquile varieties -- varieties consisting of power series in a single variable

TL;DR

This paper explores the geometry of arquile varieties, which are solution sets of equations in power series, revealing that their singularities are confined to a finite-dimensional part after stratification, blending algebraic geometry with power series analysis.

Contribution

It introduces the concept of arquile varieties and analyzes their geometric properties, especially the structure and confinement of their singularities, using a novel combination of algebraic geometry and power series techniques.

Findings

Singularities are confined to a finite-dimensional part after stratification.

Arquile varieties generalize arc spaces of algebraic varieties.

Power series solutions are key to understanding their geometry.

Abstract

Arquile varieties are zerosets of polynomial, algebraic, analytic, or formal equations f(t,y_1,...,y_m) = 0 with solutions y(t) = (y_1(t),...,y_m(t)) in affine m-space over an algebraic, convergent or formal power series ring k<t>, k{t}, or k[[t]]. As such they generalize the concept of the arc space of an algebraic variety. In the article, the geometry of arquile varieties is studied in detail. Among other things, it is shown that, after a suitable stratification, their singularities, once defined appropriately, are confined to a finite dimensional part. The main technique to do this is to combine, as is standard in the theory of arc spaces, tools from algebraic geometry and commutative algebra with the additional knowledge that the points of arquile varieties are not just abstract objects (as they are in classical algebraic and analytic geometry) but concrete power series having their proper series expansion.