Contents

# Contents

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

## Motivation

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

## 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

$(Sch/k)^{op}\to Set\,\,\,\,\,\,\,\,\,\, Y\mapsto (Sch/k) (Y\times_k k[t]/t^{m+1},X)$

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).

## Properties

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

$(Sch/k) (Y,X_\infty) \cong (ind-Sch/k) (Y\hat\times_{Spec k} Spec k[[t]],X)$

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

## Literature

Early ideas appeared in

• J. Nash Jr., Arc structure of singularities, Duke Math. J., 81 (1995), 31–38.

and its appearance in motivic integration stems from

• M. Kontsevich, lecture on motivic integration, Orsay, December 7, 1995.

For basic lectures see

• M. Mustaţǎ, Spaces of arcs in birational geometry, pdf
• M. Popa, 571 Ch. 5. Jet schemes and arc spaces, pdf

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:

• Clemens Bruschek, Hussein Mourtada, Jan Schepers, Arc spaces and the Rogers–Ramanujan identities, The Ramanujan Journal 30:1 (2013) 9-38

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