Points in linear general position in a section of a curve in $\mathbb P^{n}(\mathbb C)$

Question

Given a natural number $p\in \mathbb N \setminus{0}$, let $X\subseteq\mathbb P^{g+p+1}(\mathbb C)$ be a curve of genus $p$ and degree $2g+p+1$, and $Y\subseteq \mathbb P^{g+p}$ a general hyperplane section of $X$. Then $Y$ consists in a set of $2g+p+1$ points in linear general position: why are these points in linear general position?

Answer 1

This follows from the result below, known as General Position Theorem.

General Position Theorem. Let $C \subset \mathbb{P}^r$, $r>2$, be an irreducible, non-degenerate, possibly singular curve of degree $d$. Then a general hyperplane meets $C$ in $d$ points any $r$ of which are linearly independent.

For a proof you can see Arbarello, Cornalba, Griffiths, Harris: Geometry of Algebraic Curves I, p. 109 (Chapter III).