While for finite $m$, $m$-jets of a scheme of finite type (over an algebraically closed field of characteristic $0$) are represented by a scheme, the $\infty$-jet scheme, the (inverse) limit of $m$-jet schemes is not of finite type; this is the arc space.
The arc space (and the jet schemes) of a variety $X$ gives information about the singular locus $X_{sing}$.
Let $k$ be the algebraically closed field, $Sch/k$ the category of schemes over $k$ and $X$ an object in $Sch/k$. The presheaf
is representable by a $k$-scheme of finite type $X_m$ the $m$-jet scheme. For $s\geq 1$, the canonical maps $k[t]/t^{m+1}\to k[t]^{m+s+1}$ induces maps $(Sch/k) (Y\times_k k[t]/t^{m+s+1},X)\to (Sch/k)(Y\times_k k[t]/t^{m+1},X)$, what is $(Sch/k) (Y,X_{m+1})\to (Sch/k) (Y, X_m)$ hence also on representing objects $X_{m+1}\to X_m$. The limit is the arc space $X_\infty = lim_m X_m$ of $X$ and it comes along with natural projections $X_\infty\to X_m\to X$ (under some assumptions each of the maps is locally trivial).
If $X$ is a scheme of finite type over $k$ then there is a bijection
natural in $Y$ in $Sch/k$, where $Y\hat\times_k k[[t]]$ is the formal completion of $Y$ along subscheme $Y\times_{Spec k} \{0\}$.1
Early ideas appeared in
and its appearance in motivic integration stems from
For basic lectures see
Surveys:
Jan Denef, Francois Loeser, Geometry on arc spaces of algebraic varieties, Proceedings of 3rd ECM, Barcelona, July 10-14, 2000, math.AG/0006050
L. Ein, M. Mustaţǎ, Jet schemes and singularities, Algebraic geometry- Seattle 2005, 505–546, Proc. Sympos. Pure Math. 80, Part 2, Amer. Math. Soc., Providence, RI, 2009 MR2483946
Tommaso de Fernex, The space of arcs of an algebraic variety, arxiv/1604.02728
On connections to combinatorics and representation theory:
Other papers
J. Denef, F. Loeser, Germs of arcs on singular algebraic varieties and motivic integration, Invent. Math. 135 (1999), 201–232.
S Ishii, J Kollár, The Nash problem on arc families of singularities, Duke Math. J., 120 (2003) 601–620 math.AG/0207171
Shihoko Ishii, The arc space of a toric variety, doi arxiv/0312324
L Ein, R Lazarsfeld, M Mustaţǎ, Contact loci in arc spaces, Comput. Math. and math.AG/0303268
M Mustaţǎ, Jet schemes of locally complete intersection canonical singularities, with an appendix by David Eisenbud and Edward Frenkel, Invent. Math., 145 (2001) 397–424; Singularities of pairs via jet schemes, J. Amer. Math. Soc., 15 (2002) 599–615
Cobo Pablos, H. and González Pérez, Pedro Daniel (2012) Motivic Poincaré series, toric singularities and logarithmic Jacobian ideals. Journal of algebraic geometry, 21 (3) 495-529 pdf
Dave Anderson, Alan Stapledon, Arc spaces and equivariant cohomology, Transformation Groups 18:4 (2013) 931-969
J. Nicaise, Arcs and resolution of singularities, Manuscr. Math. 116: pp. 297-322 (2005)
W. Veys, Arc spaces, motivic integration and stringy invariants, in: Singularity theory and its applications, Adv. Stud. Pure Math. 43, Math. Soc. Japan, Tokyo (2006) 529-572
See also Corollary 4.4 in
Cambridge J. Math. 4: 4 (2016) 403-461 doi arXiv:404.7483
A formal version (ind-scheme) of free loop space for a complex algebraic variety containing the Kontsevich-Denef-Loeser arc scheme is studied in
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