Grid-free linear hypergraphs via Cayley-Bacharach

We give a new construction showing that for every $r\ge 3$, there exists an $r$-uniform linear hypergraph on $n$ vertices with ${\mathrm{\Theta }}_r(n^2)$ edges and no copy of the $r\times r$ grid. This complements the works of Füredi–Ruszinkó, Glock–Joos–Kim–Kühn–Lichev, Delcourt–Postle for $r\ge 4$, as well as the subsequent constructions of Gishboliner–Shapira and Solymosi for the case $r=3$.