Arquile varieties  varieties consisting of power series in a single variable
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 mspace 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.
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2110.08236
 Bibcode:
 2021arXiv211008236H
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Commutative Algebra;
 Mathematics  Complex Variables
 EPrint:
 To appear in Forum Math:Sigma. 41 pages, including table of symbols