Definition
If \(\mathcal F\) is a coherent $\mathcal O_X$-module we get an associated cone
\begin{align*} C(\mathcal F) = \Spec \Sym (\mathcal F). \end{align*}For any $X$-scheme \(T\), i.e. a scheme \(T\to X\), we then get
\begin{align*} C(\mathcal F)(T) &:= \Hom_{\text{Sch}/X}(T, C(\mathcal F)) \\ &= \Hom_{\mathcal O_{X}\text{-alg}}(\Sym \mathcal F, f_{*}\mathcal O_{T}) \\ &= \Hom_{\mathcal O_{X}\text{-mod}}(\mathcal F, f_{*}\mathcal O_{T}) \\ &= \Hom_{\mathcal O_{T}\text{-mod}}(f^*\mathcal F, \mathcal O_{T}) \end{align*}where the second-to-last line follows from the universal property of \(\Sym(-)\) and the final line follows from adjunction. This means that \(C(\mathcal F)\) is a group scheme over \(X\); the functor of points of \(C(\mathcal F)\) lands in \(\text{Grp}\) as demonstrated above. We call such a cone an abelian cone.
Note that if \(E\) is a geometric vector bundle over \(X\), then \(E = C(\mathscr E^\vee)\) where \(\mathscr \epsilon\) is the coherent $\mathcal O_X$-module of sections of \(E\) and \(\mathscr E^\vee\) is its dual.