A Note on Injectivity of Frobenius on Local Cohomology of Hypersurfaces

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Let k be a field of characteristic p > 0 such that [k:kᵖ] < ∞ and let f ∈ R = k[x₀,..., xₙ] be homogeneous of degree d. We obtain a sharp bound on the degrees in which the Frobenius action on Hⁿₘ(R/fR) can be injective when R/fR has an isolated non-F-pure point at m. As a corollary, we show that if (R/fR)ₘ is not F-pure then R/fR has an isolated non-F-pure point at m if and only if the Frobenius action is injective in degrees ≤ -n(d-1).