# Global generation of a vector bundle

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!

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 $$x$$.
• I'm sorry, I don't understand where the elliptic curve lives. Do you mean that the base locus of $W$ is the image of the elliptic curve under the blow-up morphism? Moreover, why is the elliptic curve passing through $x$ the base locus? Commented Jul 19, 2023 at 12:32
• Right, you should take the image of the elliptic curve under the blow-up morphism (let me cal it $f$). The point is that $|W|$ is the image of the linear system $f^*(-K_X) - E$, where $E$ is the exceptional divisor, whic in its turn is equal to the preimage of the linear system of lines on $\mathbb{P}^2$. The base locus of lines passing through a point is that point, the base locus of the preimage is the fiber over that point, and the base locus of the corresponding subsystem of $W$ is the image of this fiber under $f$. Commented Jul 19, 2023 at 19:16