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In their original paper Green and Griffiths remark that there is a differentiation operation on their jet bundles: $$ (-)' : \mathcal{J}_{k,m} \to \mathcal{J}_{k+1,m+1} $$ Which they define on p.47 ...

Let $A$ and $B$ be $R$-algebras. A Hasse-Schmidt $m$-derivation $D : A \to B$ is a tuple $(D_0, D_1, \dots, D_m)$ of $R$-linear maps $A \to B$ satisfying the generalized Leibniz law, $$ D_k(xy) = \...

(This was posted to https://math.stackexchange.com/q/4244260/799193 where it did not receive an answer.) Let $X \subseteq \mathbb{C}^n$ be an affine variety defined by $f_i(x_1, \ldots, x_n)=0, 1 \le ...

Let $X$ be a smooth variety. Because $\mathcal{O}_X$ admits a canonical connection $\mathrm{d} : \mathcal{O}_X \to \Omega_X$ the sequence, $$ 0 \to \Omega_X \to J^1(\mathcal{O}_X) \to \mathcal{O}_X \...

In this article "Introduction to Jet Schemes and Arc Spaces" S. Ishii introduces the spaces of $m$-jets: Let $X$ be a variety over algebraically closed field $k$. The space $X_m$ of $m$-jets ...