nLab
arc space

Idea

While for finite mm, mm-jets of a scheme of finite type (over an algebraically closed field of characteristic 00) are represented by a scheme, the mm-jet scheme, the (inverse) limit of mm-jet schemes is not of finite type; this is the arc space.

Motivation

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

Definition

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

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

is representable by a kk-scheme of finite type X mX_m the mm-jet scheme. For s1s\geq 1, the canonical maps k[t]/t m+1k[t] m+s+1k[t]/t^{m+1}\to k[t]^{m+s+1} induces maps (Sch/k)(Y× kk[t]/t m+s+1,X)(Sch/k)(Y× kk[t]/t m+1,X)(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)(Sch/k)(Y,X m)(Sch/k) (Y,X_{m+1})\to (Sch/k) (Y, X_m) hence also on representing objects X m+1X mX_{m+1}\to X_m. The limit is the arc space X =lim mX mX_\infty = lim_m X_m of XX and it comes along with natural projections X X mXX_\infty\to X_m\to X (under some assumptions each of the maps is locally trivial).

Properties

If XX is a scheme of finite type over kk then there is a bijection

(Sch/k)(Y,X m)(indSch/k)(Y×^ SpeckSpeck[[t]],X) (Sch/k) (Y,X_m) \cong (ind-Sch/k) (Y\hat\times_{Spec k} Spec k[[t]],X)

natural in YY in Sch/kSch/k, where Y×^ kk[[t]]Y\hat\times_k k[[t]] is the formal completion of YY along subscheme Y× Speck{0}Y\times_{Spec k} \{0\}.1

Literature

Related nnLab entries include singularity, loop space, jet space, motivic integration

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

For surveys, see

  • 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

There are connections to combinatorics and representation theory, see

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

A survey

  • Tommaso de Fernex, The space of arcs of an algebraic variety, arxiv/1604.02728

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). pp. 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

Last revised on April 12, 2016 at 15:38:39. See the history of this page for a list of all contributions to it.