Manifolds whose tangent spaces have a special behavior

Question

Consider an $n$-dimensional complex manifold $M\subset\mathbb{C}^N$ and let

$$f:\mathcal{U}\subset\mathbb{C}^n\rightarrow \mathcal{V}\subset M\subset\mathbb{C}^N$$

be a local parametrization of $M$.

Assume that for all $p\in\mathcal{U}$ we have that $f(p)$ is a linear combination of $\frac{\partial f}{\partial x_1}(p),\dots, \frac{\partial f}{\partial x_n}(p)$.

Is then $M\subset\mathbb{C}^N$ an affine subspace of $\mathbb{C}^N$?

In general, does there exist a term for a point $q = f(p)\in M$ such that $f(p)$ is a linear combination of $\frac{\partial f}{\partial x_1}(p),\dots, \frac{\partial f}{\partial x_n}(p)$?

Answer 1

The answer to the first question is No.

The assumption you made is equivalent to stating that for every $q\in M$ that the vector $q\in T_qM$.

This is satisfied whenever $M$ is a portion of a cone, which need not be affine.