On codimension two subvarieties of 𝑃⁵ and 𝑃⁶

We prove the following: Theorem. Let X ⊂ P 5 X\subset \mathbf {P}^5 be a smooth, subcanonical threefold. If h 0 ( I X ( 4 ) ) ≠ 0 h^0(\mathcal {I}_X(4))\ne 0 , then X X is a complete intersection. Let X ⊂ P 6 X\subset \mathbf {P}^6 be a smooth, codimension two subvariety, if h 0 ( I X ( 5 ) ) ≠ 0 h^0(\mathcal {I}\!_X(5))\!\ne 0 or deg ⁡ ( X ) ≤ 73 \operatorname {deg}(X)\le 73 , then X X is a complete intersection. This improves, for 5 ≤ n ≤ 6 5\le n\le 6 , earlier results on Hartshorne’s conjecture for codimension two subvarieties of P n \mathbf {P}^n .