We know that any smooth projective curve can be embedded (closed immersion) in $\mathbb{P}^3$. By definition a projective scheme over $k$ admits an embedding into some $\mathbb{P}^n$. Can we create an upper bound for the $n$ required (perhaps by strengthening the hypotheses) necessary to create an embedding of a smooth projective dimension $k$ scheme into $\mathbb{P}^n$ much like Whitney's theorem tells us we can embed an $n$ dimensional manifold in $\mathbb{R}^{2n}$?



Is there a Whitney theorem type theorem for projective schemes?
Over an algebraically closed field, any projective smooth variety of dimension $n$ can be embedded in $\mathbb P^{2n+1}$. This is elementary and can be found in Shafarevich's Basic Algebraic Geometry, Chapter II, §5.4 .
Of course specific varieties might be embedded in projective spaces of lower dimension.


For an abelian variety however we have a very satisfying complete description of the situation:


For $n=1$, we can embed any abelian (=elliptic) curve in $\mathbb P^{2}$.


For $n=2$, some abelian surfaces but not all of them can be embedded in $\mathbb P^{4}$.
The others can only be embedded in $\mathbb P^{5}$. This is due to Horrocks-Mumford.


For $n\geq 3$, no abelian variety of dimension $n$ can be embedded in $\mathbb P^{2n}$. They can only be embedded in $\mathbb P^{2n+1}$. (This theorem was proved by Van de Ven.)


**Summary and references** (added later) $\quad$ Over an algebraically closed field every projective smooth variety of dimension $n$ can be embedded in $\mathbb P^{2n+1}$. The embedding dimension $2n+1$ is sharp in the sense that for every $n$ there is a projective smooth variety of dimension $n$ not embeddable in $\mathbb P^{2n}$. 


[For $n=1$ the sharpness is due to the fact that smooth curves do not embed in $\mathbb P^{2}$ unless their genus is of the form $(d-1)(d-2)/2$. For $n\geq 2$ the sharpness is due to the discussion of abelian varieties above]


G. Horrocks and D. Mumford. A rank 2 vector bundle on P4 with
15,000 symmetries. Topology 12 (1973), 63-81


A. Van de Ven. On the embedding of abelian varieties in projective spaces. Ann. Mat. Pura Appl. (4), 103:127–129, 1975.


