The Joints Problem in $\mathbb{R}^n$

We show that given a collection of $A$ lines in $\mathbb{R}^n$, $n\geq2$, the maximum number of their joints (points incident to at least $n$ lines whose directions form a linearly independent set) is $O(A^{n/(n-1)})$. An analogous result for smooth algebraic curves is also proven.