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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!

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1 Answer 1

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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 $x$.

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  • $\begingroup$ 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? $\endgroup$
    – MaryMoon
    Commented Jul 19, 2023 at 12:32
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    $\begingroup$ 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 $f$. $\endgroup$
    – Sasha
    Commented Jul 19, 2023 at 19:16

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