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

@article{Hauser2021ArquileV, title={Arquile Varieties – Varieties Consisting of Power Series in a Single Variable}, author={Herwig Hauser and Sebastian Woblistin}, journal={Forum of Mathematics, Sigma}, year={2021}, volume={9} }

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…

## One Citation

Embedding codimension of the space of arcs

- Mathematics
- 2020

We introduce a notion of embedding codimension of an arbitrary local ring, establish some general properties, and study in detail the case of arc spaces of schemes of finite type over a field.…

## References

SHOWING 1-10 OF 51 REFERENCES

On the formal arc space of a reductive monoid

- Mathematics
- 2016

Let $X$ be a scheme of finite type over a finite field $k$, and let ${\cal L} X$ denote its arc space; in particular, ${\cal L} X(k)=X(k[[t]])$. Using the theory of Grinberg, Kazhdan, and Drinfeld on…

The minimal formal models of curve singularities

- Mathematics
- 2017

Let k be a field. We introduce a new geometric invariant, namely the minimal formal models, associated with every curve singularity (defined over k). This is a noetherian affine adic formal k-scheme,…

The Drinfeld-Grinberg-Kazhdan Theorem for formal schemes and singularity theory

- Mathematics
- 2017

Let k be a field. In this article, we provide an extended version of the Drinfeld-Grinberg-Kazhdan Theorem in the context of formal geometry. We prove that, for every formal scheme V topologically of…

Germs of arcs on singular algebraic varieties and motivic integration

- Mathematics
- 1999

We study the scheme of formal arcs on a singular algebraic variety and its images under truncations. We prove a rationality result for the Poincare series of these images which is an analogue of the…

On the Grinberg - Kazhdan formal arc theorem

- Mathematics
- 2002

Let X be an algebraic variety over a field k, and L(X) be the scheme of formal arcs in X. Let f be an arc whose image is not contained in the singularities of X. Grinberg and Kazhdan proved that if k…

The classical Artin approximation theorems

- Mathematics
- 2017

The various Artin approximation theorems assert the existence of power series solutions of a certain quality Q (i.e., formal, analytic, algebraic) of systems of equations of the same quality Q,…

Isosingular loci and the Cartesian product structure of complex analytic singularities

- Mathematics
- 1978

Let X be a (not necessarily reduced) complex analytic space, and let V be a germ of an analytic space. The locus of points q in X at which the germ Xq is complex analytically isomorphic to V is…

The Basic Theory of Power Series

- Mathematics
- 1993

I Power Series.- 1 Series of Real and Complex Numbers.- 2 Power Series.- 3 Ruckert's and Weierstrass's Theorems.- II Analytic Rings and Formal Rings.- 1 Mather's Preparation Theorem.- 2 Noether's…

Divisorial Valuations via Arcs

- Mathematics
- 2007

This paper shows a finiteness property of a divisorial valuation in terms of arcs. First we show that every divisorial valuation over an algebraic variety corresponds to an irreducible closed subset…

Embedding codimension of the space of arcs

- Mathematics
- 2020

We introduce a notion of embedding codimension of an arbitrary local ring, establish some general properties, and study in detail the case of arc spaces of schemes of finite type over a field.…