The log canonical threshold of homogeneous affine hypersurfaces

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We prove that if Y is a hypersurface of degree d in Pⁿ with isolated singularities, then the log canonical threshold of (Pⁿ,Y) is at least minn/d,1. Moreover, if d is at least n+1, then we have equality if and only if Y is the projective cone over a (smooth) hypersurface in Pⁿ⁻¹. In the case when Y is a hyperplane section of a smooth hypersurface in Pⁿ⁺¹, Cheltsov and Park have proved that Y has isolated singularities and they have obtained the above lower bound for the log canonical threshold. Moreover they made the conjecture about the equality case (for d=n+1) and they proved that the conjecture follows from the Log Minimal Model Program. The purpose of this note is to give an easy proof of their conjecture using the description of the log canonical threshold in terms of jet schemes.