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Feb 7, 2017 at 0:55 comment added Jason Starr That is not true. For $X=Y=\text{Proj}\ k[u,v,w]$, for $F=\mathcal{O}_X^{\oplus 3}$, for $G=\mathcal{O}_X(1)$, for $\phi_1=(u,v,w)$, and for $\phi_2 = (-u,u-v,v-w)$, both $\phi_1$ and $\phi_2$ are surjective, but $\phi_1+\phi_2$ has image $\mathcal{I}\cdot G$, where $\mathcal{I}$ is the ideal sheaf of the closed point of the homogeneous prime ideal $\langle u,v\rangle$.
Feb 7, 2017 at 0:51 answer added user104547 timeline score: 1
Feb 7, 2017 at 0:28 history asked user43198 CC BY-SA 3.0