Jet Schemes
Let \(X\) be a scheme of finite type over \(k\), the jet scheme \(J_m(X)\) is characterized by the following identification of functor of points
\[\mathrm{Hom}(\mathrm{Spec}(A), J_m(X)) \cong \mathrm{Hom}(\mathrm{Spec}(A[t]/(t^{m+1})), X)\]
One can view this as an adjoint pair of functors, and the ordinary points of the jet scheme corresponds to thickened points of the original scheme
\[ J_m(X)(\mathrm{Spec}(k)) \cong X(\mathrm{Spec}(k[t]/(t^{m+1}))).\]
For example one can imagine \(J_1(X)\) as the set of tangent vectors, elements in the tangent bundle \(TX\). When \(m=0\), the jet scheme reduces to the original scheme \(J_0(X)=X\).
There is a canonical projection map when \(n>m\),
\[\pi_{n,m}: J_n(X) \to J_m(X)\]
induced by the inclusion map \(k[t]/(t^{m+1}) \to k[t]/(t^{n+1})\) (which sets higher order terms zero). This is a direct system, so \(J_n(X)\) forms a projective system and we can take the limit
\[J_\infty(X) = \varprojlim J_n(X)\]
which is thought as arc space of \(X\), whose functor of points is obviously the limit
\[\mathrm{Hom}(\mathrm{Spec}(A),J_\infty(X)) = \mathrm{Hom}(\mathrm{Spec}(A[[t]]),X)\]
where \(A[[t]]\) is the formal power series ring over \(A\).
The \(J_n\) is a covariant endofunctor in the category of schemes of finite type over \(k\), if \(f:X\to Y\) we would have \(f_m := J_m(f) : J_m(X)\to J_m(Y)\), defined as ‘pushforward’
\[ \gamma : \mathrm{Spec}(k[t]/(t^{m+1}))\to X \mapsto f\circ \gamma : \mathrm{Spec}(k[t]/(t^{m+1}))\to Y.\]
The functor maps are compatible with the projection maps.
Examples
Jet Schemes of Affine Spaces
Let \(X=\mathbb{A}^n_k\), we have
\[\begin{align*} J_m(\mathbb{A}^n_k) &= \mathrm{Hom}(k[x_1,\dots,x_n], k[t]/(t^{m+1})) \\ &= (k[t]/(t^{m+1}))^n \\ &\cong \mathbb{A}^{n(m+1)}_k. \end{align*}\]
The map \(k[x_1,\dots,x_n]\to k[t]/(t^{m+1})\) specifies a \(O(t^{m+1})\) formula as a function of \(t\) for each of the coordinate \(x_i\)
\[ x_i = a_{i0} + a_{i1}t + \dots + a_{im}t^m + O(t^{m+1})\]
and this defines a local order \(m\) curve, this local curve is a point in the jet scheme \(J_m(\mathbb{A}^n_k)\).
(Random thought: hey this looks like a point in a Hilbert scheme, for example at a smooth point of our variety, when two points collide, it remembers a tangent direction (but without direction, it’s a projective direction) we seem to have a point in the first jet space. Maybe if we define a \(\mathbb{C}^*\) action on the jet space, and take quotient, we will arrive at some projective jet scheme that is similar to Hilbert scheme?)
Easy Properties
If \(J_m(X)\) exists and \(U\subset X\) is open, then \(J_m(U)\) exists as well given by \(J_m(U) = \pi_m^{-1}(U)\).
If \(X\hookrightarrow Y\) is an embedding, then \(J_m(X)\hookrightarrow J_m(Y)\) is also an embedding.
Jet functor commutes with finite products,
\[J_m(X\times Y) \cong J_m(X)\times J_m(Y).\]
For a group scheme \(G\) over \(k\), the jet scheme \(J_m(G)\) is also a group scheme over \(k\). (imagine it as multiplication of two local curves)
For \(f:Y\to X\) a morphism and \(Z\subset X\) closed embedding,
\[J_m(f^{-1}Z) = f_m^{-1}J_m(Z)\]
For any etale morphism \(f:X\to Y\), the diagram
is cartesian. Intuitively this means you can lift a local curve uniquely over an etale morphism. (Does this look like the Homotopy lifting property in homotopy theory?)