I apologize if this is a simple question, but does the Grassmannian of lines in $\mathbb P\_k^3$, $\mathbb G(1,3)$, embed into $\mathbb P\_k^5$ when $k$ an algebraically closed field of characteristic $2$? 



Plucker embedding in char 2
The most general form of the Plücker embedding I know of is the following, which you can find in EGA I, 9.8: Let $S$ be an arbitrary base scheme and $\mathcal{F}$ be a quasi-coherent sheaf on $S$. Then there is a canonical morphism of $S$-schemes $\mathrm{Grass}\_n(\mathcal{F}) \to \mathbb{P}(\Lambda^n(\mathcal{F}))$. It is defined via the functors of points: If $p : T \to S$ is an $S$-scheme, then $\mathrm{Grass}\_n(\mathcal{F})(T) \to \mathbb{P}(\Lambda^n(\mathcal{F}))(T)$ maps $p^\* \mathcal{F} \twoheadrightarrow \mathcal{G}$ (for $\mathcal{G}$ locally free of rank $n$) via $\Lambda^n(-)$ to $p^\* \Lambda^n(\mathcal{F}) \twoheadrightarrow \Lambda^n(\mathcal{G})$. Using this functorial definition, it can be shown that it is a closed immersion (loc. cit. 9.8.4). I think this is far more enlightening and transparent than the description with homogeneous coordinates.


It is also possible to write down the quasi-coherent ideal in $\mathbb{P}(\Lambda^n(\mathcal{F}))$ which corresponds to the closed immersion (the EGA proof does not produce this), it is given by the so called Plücker relations. This can be proven, again, in a functorial way. So after all you don't really have to work internally in $S$, but rather above $S$. Even more is true, the whole thing can be categorified to a theorem in arbitrary cocomplete tensor categories (beyond categories of quasi-coherent algebras), basically because the arguments are just algebraic in nature. No characteristic assumptions are needed, even less an algebraically closed field. Of course varieties over an algebraically closed field have very nice properties, but here it is useful to adopt Grothendieck's relative/functorial point of view.


