math.AG — Algebraic Geometry
Point Separation for Adjoint Line Bundles
Let X be a smooth complex projective variety of dimension n and let L be an ample line bundle. Using the freeness and minimization inputs specified in the introduction, we prove that KX + mL separates distinct points for every integer m ≥ n + 2, the sharp bound predicted by Fujita’s very ampleness conjecture. Exponential-moment convergence produces point-centred minimizing valuations at both points in a common degree. Strict convexity then gives a divisor with discrepancy equality at one point and strict excess at the other. A rational perturbation on a resolution, followed by Kawamata–Viehweg vanishing and descent, gives the required sections
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