Towards a Separation Theorem of Points by Adjoint Linear Systems on Polarized Threefolds

If l 6⊂ M , let π2 : M2−→M1 be the blow-up at y and let E2 be its exceptional divisor. Then LM2 −π ∗ 2E1−E2 is nef as in (0.5), and we get a contradiction by using vanishing theorem. If l ⊂ M , there is a smooth member D of |L| such that l ⊂ D, provided n ≥ 3. Let D1 be the proper transform of D on M1. Then y ∈ D1 and [π (K + tL)−E1]D1 = [π ∗(KD + (t− 1)LD)− E1]D1 is spanned at y by the induction hypothesis on n. This yields a contradiction, as desired.