We define a class of plane curves that are close to the free divisors in terms of the local cohomology of their Jacobian algebras and such that, conjecturally, any rational cuspidal curve C is either free or belongs to this class. We prove this conjecture when the degree of C is either even or a prime power, or when the group of C is abelian.
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Dimca et al. (2018) studied this question.
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