# Zero section of quasi-coherent bundle

Let $$S$$ be a scheme and let $$\mathcal{E}$$ be a quasi-coherent $$\mathcal{O}_S$$-module. Then we can construct a graded quasi-coherent $$\mathcal{O}_S$$-algebra $$\mathscr{A}:= Sym(\mathcal{E})$$ and define the quasi-coherent bundle $$\mathbb{V}(\mathcal{E}) := Spec(Sym(\mathcal{E}))$$. we set $$Sym_0(\mathcal{E}):= \mathcal{O}_S$$ the $$0$$-degree elements and denote $$f: \mathbb{V}(\mathcal{E}) \to S$$ the structure morphism.

It's easy to show that $$\mathbb{V}(\mathcal{E})$$ represents the functor $$F:(Sch/S) \to (Set), (h:T \to S) \mapsto \Gamma(T, h^* \mathcal{E}^{\vee})$$.

this is the consequence of identifications $$Hom_S(T,\mathbb{V}(\mathcal{E})) = Hom_{(\mathcal{O}_S -Alg)}(Sym(\mathcal{E}), h_* \mathcal{O}_T)=Hom_{\mathcal{O}_S}(\mathcal{E}, h_* \mathcal{O}_T)=Hom_{\mathcal{O}_T}(h^*\mathcal{E}, \mathcal{O}_T)=\Gamma(T, h^* \mathcal{E}^{\vee})$$

the zero element $$0 \in \Gamma(T, h^* \mathcal{E}^{\vee})$$ corresponds via identifications above to zero section $$i: S \to \mathbb{V}(\mathcal{E})$$, i.e. $$f \circ i= id_S$$ (section property). going through the identifications it's easy to see that the zero section is induced by augmentation homomorphism $$Sym(\mathcal{E}) \to Sym(\mathcal{E})/Sym(\mathcal{E})_+ = \mathcal{O}_S$$ simply by "killing" the elements of grades $$\ge 1$$. this implies that the zero section $$i$$ is a closed immersion.

$$C:=\mathbb{V}(\mathcal{E})$$ is also called the "cone". we can "remove" the image of the zero section and obtain $$C^0:= C \backslash i(S)$$.

The structure morphism $$f: C \to S$$ corresponds once more by our identifications to $$id_C$$ and a homomorphism

$$u: f^*\mathcal{E} \to \mathcal{O}_C$$

of $$\mathcal{O}_C$$-modules.

Q: why is $$C^0$$ is exacly the locus, where $$u$$ is surjective? i.e. why $$C^0= C \backslash (Coker(u))$$ or equivalently $$i(S) =Supp(Coker(u))$$?

recall: $$u$$ is surjective at $$c \in C$$ if and only if $$u_c: (f^*\mathcal{E})_c \to \mathcal{O}_{C,c}$$ is surjective homomorphism of $$\mathcal{O}_{C,c}$$-modules.

P.S. 1) all previously used notations & definitions are from Görtz' and Wedhorn's Algebraic Geometry.

P.S. 2) I have already tried to ask similar question at MSE without succeeding in solving the problem.