I learned in this script on Geometry of Algebraic Curves by Joe Harris that a rational normal curve can be constructed by a nice, old-fashioned construction, called the Steiner construction (page 79). It works as follows:
Let say we have $\Lambda_1, ..., \Lambda_n \cong \mathbb{P}^{n-2} \subset
\mathbb{P}^{n} $ codimension two subspaces in
$\mathbb{P}^{n} $. Let for every $i =1,...,n$ be $\{H_t^i \}_{t \in \mathbb{P}^{1}}$
the pencil of hyperplanes containing the plane $\Lambda_i$
with
additional assumption that for every $t_0 \in \mathbb{P}^{1}$ the planes
$H_{t_0}^1,...H_{t_0}^n$ are independent, i.e., intersect in a unique point $p(t_0)$.
Consider the union of intersections of these hyperplanes:
$$ C:= \bigcup_{t \in \mathbb{P}^{1}} H_t^1 \cap ... \cap H_t^n $$
Claim: $C \subset \mathbb{P}^{n} $ is a rational normal curve.
Here I tried to prove it directly by tour de force, ie solving simultaneously the linear system consisting of $n$ linear equations
$$ \lambda L_{1, i} + \mu L_{2, i} = a_{i,Z_0} (t)Z_0+ a_{i,Z_1} (t)Z_1 + ... + a_{i, Z_d} (t)Z_d $$
associated to vanishing sets of $H_t^i$ where $t=(\lambda: \mu)$ and $L_{j,i}$ linear forms in $Z_0,..., Z_n$. The final solution was that if we define the matrix $A \in \operatorname{Mat}_{n+1}(K[\lambda_0, \lambda_1])$ such that it's $i$-th row equals $(a_{i,Z_0} (t), a_{i,Z_1} (t), ... a_{i, Z_d} (t) )$ and the $n+1$ row is the zero row, then one can check that the solution is the last coloumn of the adjugate matrix $R$ to $A$, that's the unique matrix with $AR=RA= det(A) \cdot I_{n+1} 0$. Ok, but now my
Question is if it is also possible to show that $C$ is rational normal curve by a coordinate free argument using intersection count. Namely it is known that ration normal curves in $\mathbb{P}^{n} $ are exactly irreducible, smooth curves of degree $n$.
So our job is to verify that $C:= \bigcup_{t \in \mathbb{P}^{1}} H_t^1 \cap ... \cap H_t^n$ is irreducible, smooth of degree $n$.
It is irreducible since the map $\mathbb{P}^{1} \to C, t \mapsto p(t)$ is continuous. How can we show smoothness and degree $n$? Firstly is it non-degenerated (see also this question which dedicated the non-degeneracy of $C$)?
For degree we have to show that for general hyperplane $H \subset \mathbb{P}^{n} $ the intersection $C \cap H$ consists of exactly $n$ points. Could somebody help?