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?



How can I get the scheme-theoretic support of coherent sheaf on a ruled surface with linear Hilbert bipolynomial ax+by+c?
**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.


