Smallest projective subspace containing a degree $d$ curve Is it true that the smallest projective subspace containing a degree $d$ curve inside $\mathbb{P}^n$ has dimension at most $d$? If not, is there any bound on the dimension? Generalization to varieties?
For $d=1$ this is obvious. I think for the case that the curve is an embedding of $\mathbb{P}^1$ this is also true: Suppose the embedding is given by $n$ degree $d$ homogeneous polynomials $f_0,\dots,f_n$. For each $0\leq i\leq d$, let $p_i=(c_{0,i},\dots,c_{n,i})$ where $c_{j,i}$ is the coefficient of $x^iy^{d-i}$ in $f_j$ (or we ignore $p_i$ if all $c_{j,i}$ are zero). Then the curve is contained in the projective subspace spanned by all $p_i$.

I think your observation is correct for curves.
Given a curve $C$ in $\mathbb{P}^n$ satisfying $C$ is not contained in any projective subspace of $\mathbb{P}^n$, WLOG we may assume $C$ is irreducible. Let $\tilde C$ be the normalization of $C$, then we have a regular map $\phi: \tilde C\rightarrow \mathbb{P}^n$ which is an embedding outside a finite subset of $\tilde C$.
Since $C$ is not contained in any projective subspace, $\phi$ is the map induced by the linear series $L$ of all hyperplane divisors. It is clear that $\dim L=\dim \mathbb{P}^n=n$, so for any hyperplane divisor $D$, we have $h^0(\tilde C,D)\geq \dim L+1=n+1$. On the other hand, $h^0(\tilde C,D)\leq \deg(D)+1$ since $D$ is an effective divisor, so $\deg C=\deg D\geq n$.
Any generalizations to higher dimensional varieties?
Guess: for a $k$-dimensional variety $X$ embedded in $\mathbb{P}^n$, if $X$ is not contained in any projective subspace, then $\deg X\geq n-k+1$.
