If p is a nonsingular point, then we can define a tangent vector as an equivalence class of (nonsingular) curves, just as in the differentiable case. In fact, being nonsingular is equivalent to every tangent vector being the speed* of a curve. In this case, the intersection of all curves in the equivalence class is a zero-dimensional closed subscheme with a one-dimensional tangent space, isomorphic to the spectrum of the dual numbers described above by Martin Brandenburg.

*Not sure this is the right term, I didn't learn differential topology in English.

Warning: I'm thinking of schemes of finite type over an algebraically closed field K, and by point I mean a K-valued point.

If p is a nonsingular point, then we can define a tangent vector as an equivalence class of (nonsingular) curves, just as in the differentiable case. In fact, being nonsingular is equivalent to every tangent vector being tangent to a curve. In this case, the intersection of all curves in the equivalence class is a zero-dimensional closed subscheme with a one-dimensional tangent space, isomorphic to the spectrum of the dual numbers described above by Martin Brandenburg.

Warning: I'm thinking of schemes of finite type over an algebraically closed field K, and by point I mean a K-valued point.