nLab jet scheme

Idea and definition

For finite $m$ this is a version of $m$-jet space in algebraic geometry.

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.

An analogue of the $\infty$-jet space is the arc space.

References

Readable introduction is in

• M. Popa, 571 Ch. 5. Jet schemes and arc spaces, pdf

• A. Beilinson, V. Drinfeld, Chiral Algebras

• 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

• L Ein, M Mustaţǎ, T Yasuda, Jet schemes, log discrepancies and inversion of adjunction, Invent. Math., 153 (2003) 519–535 math.AG/0209392

• 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