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Arquile varieties are zerosets of polynomial, algebraic, analytic, or formalequations f(t,y_1,...,y_m) = 0 with solutions y(t) = (y_1(t),...,y_m(t)) inaffine 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 analgebraic variety.  In the article, the geometry of arquile varieties is studied in detail. Amongother things, it is shown that, after a suitable stratification, theirsingularities, once defined appropriately, are confined to a finite dimensionalpart. The main technique to do this is to combine, as is standard in the theoryof arc spaces, tools from algebraic geometry and commutative algebra with theadditional knowledge that the points of arquile varieties are not just abstractobjects (as they are in classical algebraic and analytic geometry) but concretepower series having their proper series expansion.

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