We know in differential geometry, given a $C^k$ manifold for $k>1$, the tangent space at a point in this manifold is parametrized by curves passing through this point modulo certain equivalence relation. The tangent space is given by the velocity vectors to these curves at this point.

Intuitively, I would think this should be true even for Zariski topology in the case $X$ is a smooth projective variety. In particular, we could take a tangent vector $t$ and associate to this the subscheme $Y_t$ of $X$ obtained by intersecting ~~all~~ some subschemes of $X$ which has $t$ as an element in its tangent space. The question is whether we can somehow ensure that $Y_t$ is $1$-dimensional?