I'm reading Lazarfeld's book «Positivity in Algebraic Geometry I» and I'm stuck on the construction of a big and nef divisor on a variety $X$ such that its canonical ring/algebra $R(X,D) = \bigoplus_{n \geq 0} H^0(X,nD)$ is not finitely generated. Here is the text :
Start with a smooth cubic curve $C_0 \subset \mathbb P^2$ and denote by $l$ the hyperplane class on $\mathbb P^2$. Choose twelve general points $P_1, \dotsc, P_{12} \in C_0$ in such a way that the line bundle $$\eta := \mathcal O_{C_0}(P_1 + \dotsc + P_{12} - 4l) \in \text{Pic}^0(C_0)$$ is a non-torsion class ($C_0$ being an elliptic curve, this is always possible). Now take $\mu : X \to \mathbb P^2$ to be the blowing-up of $\mathbb P^2$ at the chosen points, with exceptionial divisor $E = \sum_{i=1}^{12} E_i$ ($E_i$ being the component above $P_i$). Write $H = \mu^* l$ for the pull-back of the hyperplane class on $\mathbb P^2$ and let $C \in \vert 3H - E\vert$ be the proper transform of $C_0$. We consider the divisor $$D = 4H - E$$ Note that $D \sim H + C$ (where $\sim$ means linearly equivalent) and consequently $D$ is big. Observing that $\mathcal O_C(D) = \eta^*$ is nef and that $D-C$ is free, one sees that $D$ is also nef. We assert that for every $m \geq 1$, the linear series $\vert mD \vert$ contains $C$ in its base-locus, whereas $\vert mD - C \vert$ is free. This implies that $R(X, D)$ is not finitely generated (finite generation requires that the multiplicity of a base-curve in $\vert mD\vert$ must go to infinity with $m$, see Theorem 2.3.15). The fact that $C \subset \text{Bs} \vert mD\vert$ is clear since $\mathcal O_C(mD) = \eta^{\otimes-m}$ has no sections (because of the Riemann-Roch theorem and the fact that $\eta$ is a non-torsion class). On the other hand : $mD - C \sim (m-1)D + H$, and using the exact sequence $$0 \to \mathcal O_X\big( (m-1)D + H - C \big) \xrightarrow{\cdot C} \mathcal O_X\big( (m-1)D + H \big) \to \mathcal O_C \big( (m-1)D + H \big) \to 0$$ one shows by induction on $m$ that $mD - C$ is free (observe that the term on the right is is a line bundle of degree $3$ on the elliptic curve $C$, hence free).
The last sentence is unclear for me, I don't see how to conclude. I think that I can show by induction that $$H^1\Big( X, O_X\big( (m-1)D + H - C \big) \Big) = 0$$ and using the surjection on the global sections, I can say that $C$ is not in the base-locus of $\vert(m-1)D + H\vert = \vert mD - C \vert$ because the line bundle on the right of the short exact sequence is basepoint-free ; can you help me prove that $X \backslash C$ is also not included in it? Thank you.
By induction you know that $|(m−1)D−C|$ is free, so the base locus of $|(m−1)D−C|$ is at most $C$. Since $H$ is free, the base locus of $|(m−1)D+H|$ is also at most $C$.