We define the minimal plus-one generated curves and prove a result explaining why they are the closest relatives of the free curves, after the nearly free curves. Then we look at the projective closures of the general and of the special fibers of some Brian{c}on-type polynomials constructed by E. Artal Bartolo, Pi. Cassou-Nogu\`es and I. Luengo Velasco. They yield new examples of free, nearly free or minimal plus-one generated curves. This leads us to ask the natural question if such a result holds for any Brian{c}on-type polynomials.
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Dimca et al. (2024) studied this question.
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