Primitive arcs on curves
The arc space \(X_\infty\) of an algebraic variety \(X\) carries important information about its singular locus \(X_{\mathrm{sing}}\), in particular about its numerical invariants, and about the essential divisors over \(X\). The formal neighborhood of an arc \(\alpha\in X_\infty\) could be seen as a space parametrizing its formal deformations in the space of arcs. The theorem of \textit{V. Drinfeld} [``On the Grinberg-Kazhdan formal arc theorem'', \url{arXiv:math/0203263}] and \textit{M. Grinberg} and \textit{D. Kazhdan} [Geom. Funct. Anal. 10, No. 3, 543--555 (2000; Zbl 0966.14002)] describes the formal neighborhood of any non-constant closed arc in \(X_\infty\), whose image is not entirely contained in \(X_{\mathrm{sing}}\). This result has been further developed in author's papers [the author and \textit{D. Bourqui}, ``The Drinfeld-Grinberg-Kazhdan theorem is false for singular arcs'', J. IMJ (to appear); ``The Drinfeld-Grinberg-Kazhdan theorem for formal schemes and singularity theory'', (to appear); J. Singul. 16, 130--140 (2017; Zbl 1366.14019)]. In the article under review is introduced the notion of a primitive arc at a point on curve (which is unique if exists). It is used to investigate the smoothness at closed point \(x\) of analytically irreducible curve \(X\) in terms of the formal neighborhood of the constant arc \(\alpha_x\in X_\infty\) at \(x\). The main result of the article provides criteria, each one equivalent to the existence of a primitive arc at \(x\). Among them are the following: the smoothness of the curve at x, and, that the formal neighborhood of \(\alpha_x\) is isomorphic to \(\mathbb D^\infty=Spf kX_i,i\in\mathbb N\). By taking the normalization morphism \(\nu:\overline X\to X\), we have induced morphism \(\hat\nu\) between the formal neighborhood of \(\alpha_{\bar x}\in\bar X_\infty\), where \(\bar x=\nu^{-1}(x)\), and the formal neighborhood of \(\alpha_x\in X_\infty\). The theorem claims further that the existence of a primitive arc at \(x\) is equivalent to \(\hat\nu\) being an isomorphism. This theorem can be further generalized for any geometrically unibranch integral curve and a closed point on it, by changing the base. It could be noted also, that for curve singularity \((X,x)\) the formal neighborhood of the constant arc \(\alpha_x\) represents a formal invariant, interesting to be investigated from the point of view of singularity theory.
scientific article; zbMATH DE number 6684726