Let
$ {\mathcal C} $
be an algebraic curve and c be an analytically irreducible singular point of
${\mathcal C}$
. The set
${\mathscr {L}_{\infty }}({\mathcal C})^c$
of arcs with origin c is an irreducible closed subset of the space of arcs on
${\mathcal C}$
. We obtain a presentation of the formal neighborhood of the generic point of this set which can be interpreted in terms of deformations of the generic arc defined by this point. This allows us to deduce a strong connection between the aforementioned formal neighborhood and the formal neighborhood in the arc space of any primitive parametrization of the singularity c. This may be interpreted as the fact that analytically along
${\mathscr {L}_{\infty }}({\mathcal C})^c$
the arc space is a product of a finite dimensional singularity and an infinite dimensional affine space.
We show that there exists a strong connection between the generic formal neighborhood at a rational arc lying in the Nash set associated with a toric divisorial valuation on a toric variety and the formal neighborhood at the generic point of the same Nash set. This may be interpreted as the fact that analytically along such a Nash set the arc scheme of a toric variety is a product of a finite dimensional singularity and an infinite dimensional affine space.
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