Let $X$ be a compact complex manifold (say of dimension $2$) and $L \rightarrow X $ a holomorphic line bundle. Consider the following statements:
Statement $A_0$: Given any point $p\in X$, there exists a homolorphic section $f_{00} \in H^0(X,L)$, such that $f_{00}(p) \neq 0$.
Statement $A_1$: Given any point $p\in X$, there exists two homolorphic sections $f_{10}, ~~f_{01} \in H^0(X,L)$, such that $$ f_{10}(p), ~~~f_{10}(p) =0 $$ and $\nabla f_{10}\big{|}_p$ and $\nabla f_{01}\big{|}_p$ are linearly independent in $TX_p^* \otimes L_p$.
$\textbf{Question:}$ Give a sufficient criteria on the line bundle $L \rightarrow X $ to guarantee that statement $A_0$ and $A_1$ are true. For example, does being "sufficiently ample" guarantee this?
$\textbf{Edit:} $ Statement $A_1$ was incorrectly stated before. I have also removed the remaining statements, since they were not what I wanted.
