Geodesic
Deformations of arcs and comparison of formal neighborhoods for a curve singularity
Forum of Mathematics, Sigma, Tome 11 (2023)

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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. 
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     author = {David Bourqui and Mario Mor\'an Ca\~n\'on},
     title = {Deformations of arcs and comparison of formal neighborhoods for a curve singularity},
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David Bourqui; Mario Morán Cañón. Deformations of arcs and comparison of formal neighborhoods for a curve singularity. Forum of Mathematics, Sigma, Tome 11 (2023). doi: 10.1017/fms.2023.24

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