In Mumford's GIT, the definition of level $n$ structure ($n \geq 2)$ is $2g$ sections $\{\sigma_1, \dots, \sigma_{2g}\} : S \rightarrow A$ such that two conditions hold: (i) For geometric points the sections form a basis for the $n$-torsion points and (ii) $n_A \circ \sigma_i= \varepsilon$.

My question is: Doesn't (i) imply (ii)? Corollary 6.2 implies that since the two maps are equal on geometric fibers by (i), they must be equal everywhere.



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