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I have pure sheaves of dimension 1 on a ruled surface, in paticular the Hirzebruch surface F$_e$=P($O \oplus O(-e)$) with linear Hilbert bipolynomial $P(x, y)=ax+by+c$.

A sheaf $E$ is pure of dimension d means that for all non trivial coherent subsheaves $F \subseteq E$, $\dim(F)=d$. Also, $\dim(F)$ is the dimension of support of $F$.

Then, how can I get the scheme-theoretic support?

Or where can I find the hint about this problem?

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Question: "Then, how can I get the scheme-theoretic support? Or where can I find the hint about this problem?"

Answer: If $A$ is a commutative ring and $M$ a finitely generated $A$-module, the support of $M$, denoted $Supp(M)$ is the closed subscheme $V(ann(M)) \subseteq S:=Spec(A)$. Here $ann(M) \subseteq A$ is the annihilator ideal of $M$. There is always a map $\rho: \mathcal{O}_{F_e} \rightarrow End_{\mathcal{O}_{F_e}}(F)$, and the annihilator $ann(F):=ker(\rho) \subseteq \mathcal{O}_{F_e}$ is the ideal sheaf defined as the kernel of $\rho$. If $U \subseteq F_e$ is an open subset, you get a map

$$\rho_U: \mathcal{O}_{F_e}(U) \rightarrow End_{\mathcal{O}_{F_e}}(F)(U)$$

with

$$\rho_U(s)(x):=sx.$$ You may check that $\rho$ is a well defined map of $\mathcal{O}_{F_e}$-modules. Since $F_e$ is coherent it follows $ann(F)$ is a coherent sheaf of ideals and to it corresponds a closed subscheme of $F_e$.

This defines the support $Supp(F):=V(ann(F)) \subseteq F_e$ canonically as a scheme.

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  • $\begingroup$ Thank you for the answer. I have one more question. I will edit the above question a bit. On above condition, the scheme-theoretic support is contained in |$O_{F_e}(m, n)$|. How can I get m, n? $\endgroup$
    – H.S. Kim
    Commented Jun 8, 2021 at 6:48

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