Let $X$ be a smooth projective variety of dimension $n\geq2,$ and let $A$ be an ample line bundle such that its base locus $B=\mathrm{Bs}(|A|)$ is a finite set. Let $\mathcal{E}$ be a locally free sheaf on $X,$ and fix a closed point $x\in X.$ I have the following questions.
(1) If $W=H^0(X,A\otimes\mathfrak{m}_x),$ where $\mathfrak{m}_x$ is the ideal sheaf of $x,$ is the subspace of global sections of $A$ vanishing at $x,$ the base ideal $\mathcal{J}_W$ of $W$ is the image of the evaluation map $W\otimes A^\ast\rightarrow\mathcal{O}_X$ (Positivity in Algebraic Geometry I, Def. 1.1.8). Is it true that the base scheme $Z=\mathrm{Bs}(|W|)$ is still a finite set (containing $x$)? I think it should be so because it should contain $x,$ the finite set $B$ and some other point (as in the case of Corollary V.4.5 in Hartshorne's book for cubics passing through 8 points in general positions).
(2) If one has that $H^1(X,\mathcal{E}\otimes\mathcal{J}_W)=0,$ is it true that $\mathcal{E}$ is generated by global sections at every point of $Z$? This should follow by taking the cohomology of the short exact sequence
$$
0\rightarrow\mathcal{E}\otimes\mathcal{J}_W\rightarrow\mathcal{E}\rightarrow\mathcal{E}\otimes\mathcal{O}_Z\rightarrow0
$$
which, under this assumption, gives the surjectivity of the map $H^0(X,\mathcal{E})\rightarrow H^0(X,\mathcal{E}\otimes\mathcal{O}_Z).$
(3) If (1) and (2) hold for every point $x\in X,$ does this mean that $\mathcal{E}$ is generated by global sections?
Thank you in advance!
Best Answer
No for (1). For instance, let $X$ be a del Pezzo surface of degree 1 and $A = -K_S$. Then $\mathrm{Bs}(A) = x_0$ is a single point, there is an elliptic fibration $$ \mathrm{Bl}_{x_0}(X) \to \mathbb{P}^1, $$ and $\mathrm{Bs}(|W|)$ is the elliptic curve passing through $x$.