When the base locus of V is not empty, the above discussion still goes through with \mathcal{O}_X in the direct sum replaced by an ideal sheaf defining the base locus and X replaced by the blow-up \widetilde{X} of it along the (scheme-theoretic) base locus B. Precisely, as above, there is a surjection \operatorname{Sym}((V \otimes_k \mathcal{O}_X) \otimes_{\mathcal{O}_X} L^{-1}) \to \bigoplus_{n=0}^{\infty} \mathcal{I}^n where \mathcal{I} is the ideal sheaf of B and that gives rise to
