Applications

Chambert-Loir, Antoine; Nicaise, Johannes; Sebag, Julien

doi:10.1007/978-1-4939-7887-8_7

Cited by 1 publication

(4 citation statements)

References 70 publications

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“…For , the textile topology coincides with the Zariski-topology. Compare this with the concept of Greenberg schemes developed in [CNS].…”

Section: Concepts and Resultsmentioning

confidence: 97%

“…The ring can be characterised by deformations of y over test-algebras (local algebras with nilpotent maximal ideals), and this is how Grinberg–Kazhdan and Drinfeld prove the respective factorisation result [GK, Dr, BH]. Additional information on this result can be found in [BS1, Bou, BNS, CNS, NS]. In the present text, we will establish the factorisation more generally for all deformations of y , with parameters varying in an arbitrary complete local ring S (the topology of S need not be defined by the powers of an ideal); see Theorem 10.2 for the precise formulation 12…”

Section: Concepts and Resultsmentioning

confidence: 98%

“…If is the space of arcs centred at of an algebraic subvariety X of , a point is arq.regular if and only if the image of y in X (i.e., the generic point of y ) does not lie entirely in the singular locus of X ; that is, if . This condition appears naturally in the theory of arc spaces; see, for example, [GK, Dr, DL, BS1, BS2, Bou, dFEI, CNS].…”

Section: Concepts and Resultsmentioning

confidence: 99%

“…This is quite standard: As the arc y does not lie entirely in the singular locus of X , it must lie in precisely one irreducible component. By Lemma 8.6 of [CdFD] (see also Subsection 3.4 in [CNS]), we may assume that X is irreducible. Represent X as an irreducible component of a complete intersection .…”

Section: Deformationsmentioning

confidence: 99%

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

Hauser

Woblistin

2021

Forum of Mathematics, Sigma

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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. Our results are established simultaneously for algebraic, convergent and formal power series, as well as convergent power series with prescribed radius of convergence. The key technical tool is a linearisation theorem, already used implicitly by Greenberg and Artin, showing that analytic maps between power series spaces can be essentially linearised by automorphisms of the source space. Instead of stratifying arquile varieties, one may alternatively consider formal neighbourhoods of their regular points and reprove with similar methods the Grinberg–Kazhdan–Drinfeld factorisation theorem for arc spaces in the classical setting and in the more general setting.

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“…For , the textile topology coincides with the Zariski-topology. Compare this with the concept of Greenberg schemes developed in [CNS].…”

Section: Concepts and Resultsmentioning

confidence: 97%