A note on injectivity of Frobenius on local cohomology of global complete intersections

Given a graded complete intersection ideal J=(f1,…,fc)⊆k[x0,…,xn]=S, where k is a field of characteristic p>0 such that [k:kp]<∞, we show that if S/J has an isolated non-F-pure point then the Frobenius action on top local cohomology Hmn+1−c(S/J) is injective in sufficiently negative degrees, and we compute the least degree of any kernel element. If S/J has an isolated singularity, we are also able to give an effective bound on p ensuring the Frobenius action on Hmn+1−c(S/J) is injective in all negative degrees, extending a result of Bhatt and Singh in the hypersurface case.