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We derive a numerical criterion for J-holomorphic curves in 4-dimensionalsymplectic cobordisms to achieve transversality without any genericityassumption. This generalizes results of Hofer-Lizan-Sikorav andIvashkovich-Shevchishin to allow punctured curves with boundary that generallyneed not be somewhere injective or immersed. As an application, we combine thiswith the intersection theory of punctured holomorphic curves to prove thatcertain geometrically natural moduli spaces are globally smooth orbifolds,consisting generically of embedded curves, plus unbranched multiple covers thatform isolated orbifold singularities.

Automatic Transversality and Orbifolds of Punctured Holomorphic Curves in Dimension Four