Artin-Schreier geproci configurations in projective spaces of arbitrary dimension

We construct finite geproci sets in every projective dimension and in every positive characteristic by introducing ${\mathbb{F}}_N$-Artin-Schreier configurations. In ${\mathbb{P}}^3$, we characterize exactly when such configurations are geproci: an ${\mathbb{F}}_N$-Artin-Schreier configuration on $q$ lines spanning ${\mathbb{P}}^3$ is $(q,N)$-geproci if and only if $q\le N$. We then develop a lifting construction which produces geproci sets in ${\mathbb{P}}^n$ for every $n\ge 3$.