We derive a numerical criterion for J-holomorphic curves in 4-dimensional symplectic cobordisms to achieve transversality without any genericity assumption. This generalizes results of Hofer–Lizan–Sikorav [HLS97] and Ivashkovich–Shevchishin [IS99] to allow punctured curves with boundary that generally need not be somewhere injective or immersed. As an application, we combine this with the intersection theory of punctured holomorphic curves to prove that certain geometrically natural moduli spaces are globally smooth orbifolds, consisting generically of embedded curves, plus unbranched multiple covers that form isolated orbifold singularities.
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Chris Wendl, Automatic transversality and orbifolds of punctured holomorphic curves in dimension four. Comment. Math. Helv. 85 (2010), no. 2, pp. 347–407DOI 10.4171/CMH/199