Inspired by several works on jet schemes and motivic integration, we consider an extension to singular varieties of the classical definition of discrepancy for morphisms of smooth varieties. The resulting invariant, which we call $\mathrm{𝐽𝑎𝑐𝑜𝑏𝑖𝑎𝑛𝑑𝑖𝑠𝑐𝑟𝑒𝑝𝑎𝑛𝑐𝑦}$, is closely related to the jet schemes and the Nash blow-up of the variety. This notion leads to a framework in which adjunction and inversion of adjunction hold in full generality, and several consequences are drawn from these properties. The main result of the paper...

For $m\in \mathbb{N}$, we determine the irreducible components of the $m-$th Jet Scheme of a complex branch $C$ and we give formulas for their number $N\left(m\right)$ and for their codimensions, in terms of $m$ and the generators of the semigroup of $C$. This structure of the Jet Schemes determines and is determined by the topological type of $C$.