Three-point Riemann-Roch construction of a plane cubic ($F=[1:f:g]$)

On a genus-one curve, for distinct $p,q,r$, the Riemann-Roch theorem gives $\ell (p+q+r)=3$. Choose $f$ with simple poles exactly at $p,q$ and $g$ with simple poles exactly at $q,r$. Pole comparison proves $1,f,g$ independent. Multiplying these coordinates by a vanishing local parameter extends $F$ across every pole, with images $[0:1:0]$, $[0:b:c]$, $[0:0:1]$, where $bc\ne 0$. The ten homogeneous cubic monomials evaluate in $L(3p+3q+3r)$, of dimension nine, so a nonzero homogeneous cubic vanishes on the image. The three chosen points map to the line with first coordinate zero. This connects linear systems to projective plane equations by a dimension count.