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\begin{document}

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\section*{arc space}

\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

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 $m$-[[jet scheme]], the (inverse) limit of $m$-jet schemes is not of finite type; this is the arc space.

\hypertarget{motivation}{}\subsection*{{Motivation}}\label{motivation}

The arc space (and the jet schemes) of a variety $X$ gives information about the singular locus $X_{sing}$.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

Let $k$ be the algebraically closed field, $Sch/k$ the category of schemes over $k$ and $X$ an object in $Sch/k$. The presheaf

\begin{displaymath}
(Sch/k)^{op}\to Set\,\,\,\,\,\,\,\,\,\,
 Y\mapsto (Sch/k) (Y\times_k k[t]/t^{m+1},X)
\end{displaymath}
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 \textbf{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).

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

If $X$ is a scheme of finite type over $k$ then there is a bijection

\begin{displaymath}
(Sch/k) (Y,X_m) \cong (ind-Sch/k) (Y\hat\times_{Spec k} Spec k[[t]],X)
\end{displaymath}
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

\hypertarget{literature}{}\subsection*{{Literature}}\label{literature}

Related $n$Lab entries include [[singularity]], [[loop space]], [[jet space]], [[motivic integration]]

Early ideas appeared in

\begin{itemize}%
\item J. Nash Jr., \emph{Arc structure of singularities}, Duke Math. J., 81 (1995), 31--38.

\end{itemize}
and its appearance in motivic integration stems from

\begin{itemize}%
\item [[M. Kontsevich]], lecture on motivic integration, Orsay, December 7, 1995.

\end{itemize}
For basic lectures see

\begin{itemize}%
\item M. Musta, \emph{Spaces of arcs in birational geometry}, \href{http://www.math.lsa.umich.edu/~mmustata/lectures_arcs.pdf}{pdf}
\item M. Popa, 571 Ch. 5. \emph{Jet schemes and arc spaces}, \href{http://homepages.math.uic.edu/~mpopa/571/chapter5.pdf}{pdf}

\end{itemize}
\hypertarget{for_surveys_see}{}\subsection*{{For surveys, see}}\label{for_surveys_see}

\begin{itemize}%
\item Jan Denef, Francois Loeser, \emph{Geometry on arc spaces of algebraic varieties}, Proceedings of 3rd ECM, Barcelona, July 10-14, 2000, \href{http://arxiv.org/abs/math/0006050}{math.AG/0006050}
\item L. Ein, M. Musta, \emph{Jet schemes and singularities}, Algebraic geometry- Seattle 2005, 505--546, Proc. Sympos. Pure Math. 80, Part 2, Amer. Math. Soc., Providence, RI, 2009 \href{http://www.ams.org/mathscinet-getitem?mr=2483946}{MR2483946}

\end{itemize}
There are connections to combinatorics and representation theory, see

\begin{itemize}%
\item Clemens Bruschek, Hussein Mourtada, Jan Schepers, \emph{Arc spaces and the Rogers--Ramanujan identities}, The Ramanujan Journal 30:1 (2013) 9-38

\end{itemize}
A survey

\begin{itemize}%
\item Tommaso de Fernex, \emph{The space of arcs of an algebraic variety}, \href{http://arxiv.org/abs/1604.02728}{arxiv/1604.02728}

\end{itemize}
Other papers

\begin{itemize}%
\item J. Denef, F. Loeser, \emph{Germs of arcs on singular algebraic varieties and motivic integration}, Invent. Math. 135 (1999), 201--232.
\item S Ishii, J Koll\'a{}r, \emph{The Nash problem on arc families of singularities,} Duke Math. J., 120 (2003) 601--620 \href{http://arxiv.org/abs/math/0207171}{math.AG/0207171}
\item Shihoko Ishii, \emph{The arc space of a toric variety}, \href{http://dx.doi.org/10.1016/j.jalgebra.2003.12.015}{doi} \href{http://arxiv.org/abs/math/0312324}{arxiv/0312324}
\item L Ein, R Lazarsfeld, M Musta, \emph{Contact loci in arc spaces}, Comput. Math. and \href{http://arxiv.org/abs/math/0303268}{math.AG/0303268}
\item M Musta, \emph{Jet schemes of locally complete intersection canonical singularities}, with an appendix by David Eisenbud and Edward Frenkel, Invent. Math., 145 (2001) 397--424; \emph{Singularities of pairs via jet schemes}, J. Amer. Math. Soc., 15 (2002) 599--615
\item Cobo Pablos, H. and Gonz\'a{}lez P\'e{}rez, Pedro Daniel (2012) Motivic Poincar\'e{} series, toric singularities and logarithmic Jacobian ideals. Journal of algebraic geometry, 21 (3). pp. 495-529 \href{http://www.ams.org/journals/jag/2012-21-03/S1056-3911-2011-00567-5/S1056-3911-2011-00567-5.pdf}{pdf}
\item Dave Anderson, Alan Stapledon, \emph{Arc spaces and equivariant cohomology}, Transformation Groups 18:4 (2013) 931-969
\item J. Nicaise, \emph{Arcs and resolution of singularities}, Manuscr. Math. 116: pp. 297-322 (2005)
\item W. Veys, \emph{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

\end{itemize}
See also Corollary 4.4 in

\begin{itemize}%
\item Bhargav Bhatt, \emph{Algebraization and Tannaka duality}, \href{http://arxiv.org/abs/1404.7483}{arXiv/1404.7483}

\end{itemize}


\end{document}