Canonical Projective Embeddings of the Deligne-Lusztig Curves Associated to $2A2$, $2B2$ and $2G2$

The Deligne-Lusztig varieties associated to the Coxeter classes of the algebraic groups 2A2, 2B2 and 2G2 are affine algebraic curves. We produce explicit projective models of the closures of these curves. Furthermore for $d$ the Coxeter number of these groups, we find polynomials for each of these models that cut out the ${\text{F}}_q$-points, the ${\text{F}}_{q^d}$-points and the ${\text{F}}_{q^{d+1}}$-points, and demonstrate a relation satisfied by these polynomials.