(Automatic transversality, polygon case) Suppose $v_{1},\ldots v_{n-1}$ are linearly independent first order deformation vector fields. Either $\Omega(\cdot,v_{1},\ldots v_{n-1})$ vanishes identically as a 1-form on $\Sigma$, or we must have $\deg q\geq n$, and when the equality holds then $\Omega(\cdot,v_{1},\ldots v_{n-1})$ only vanishes at corners. In this case, all first order defomation vector fields are spanned by $v_{1},\ldots v_{n-1}$, the holomorphic polygon $u:\Sigma\to X$ is an immersion up to the boundary, the obstruction of the extended linearized operator vanishes, and the moduli space is smooth at $u:\Sigma\to X$.