Let $(M^{2n},J)$ be an almost complex manifold and $(\Sigma,j)$ a closed Riemann surface. Suppose $u: \Sigma \to M$ is a simple, nonconstant, $J$-holomorphic curve. Can we prove that the set $$ Z:=\{z\in \Sigma \mid |u^{-1}(u(z))|>1\} $$ is finite?
Note: It follows from Proposition 2.5.1 in McDuff-Salamon's book "J-holomorphic Curves and Symplectic Topology" that $Z$ is countable.
Yes it must be proved in that same book, or any/every basic book, going back to the fundamental proof by Micallef--White: The critical points and the self-intersection points are isolated, as there is a local coordinate description of the map near them.
$u(z) \sim (z^m, f)\in\mathbb{C}\times\mathbb{C}^{n-1}$ with $f$ satisfying a Cauchy-Riemann equation
$du_z \sim kc_0z^{k-1}+O(|z|^{k-1})\in\mathbb{C}^n$ for critical points
