Permanence properties of $F$-injectivity
We prove that $F$-injectivity localizes, descends under faithfully flat homomorphisms, and ascends under flat homomorphisms with Cohen-Macaulay and geometrically $F$-injective fibers, all for arbitrary Noetherian rings of prime characteristic. As consequences of these results, we show that the $F$-injective locus is open on rings essentially of finite type over excellent local rings. As a geometric application, we prove that if $X$ is a smooth projective variety of dimension $r \le 5$ over an algebraically closed field of characteristic $p > 3$ embedded via a $d$-uple embedding for $d \ge 3r$, then every generic projection of $X$ is $F$-pure, and hence $F$-injective. This geometric result is the positive characteristic analogue of a theorem of Doherty.