Please, I appreciate any suggestion or reference to prove if statements 1 and 2 are true, according to the following hypothesis:
Let $M$ be a curve (compact riemann surface of genus $g \geq 2$) in a projective space $\mathbb{P}^k$, $k>3$. Let $x$ be a point in $\mathbb{P}^k\setminus M$. Then:
The tangent lines to $M$, that pass through $x$, form a sub-variety of $\mathbb{P}^k$ of dimension 2.
The lines that intersect $M$ in two points, that pass through $x$, form a sub-variety of $\mathbb{P}^k$ of dimension 3.