MathDBThe rational cuspidal curve freeness conjecture
The rational cuspidal curve freeness conjecture
Let be a plane curve. A plane curve is rational cuspidal if it is rational and all its singularities are cusps, and it is free or nearly free according to the corresponding classes of plane divisors introduced in the paper.
Rational cuspidal freeness conjecture. (i) Any rational cuspidal curve in the plane is either free or nearly free. (ii) An irreducible plane curve which is either free or nearly free is rational.
The first assertion is proved in the paper for all curves of even degree, and also under further hypotheses such as abelian complement fundamental group or prime-power degree; it remains open in general. The second assertion is presented as conjectural, with examples showing that irreducibility is essential.
Equivalent formulations 1
Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.
The rational cuspidal curve freeness conjecture
Let be a reduced plane curve. Suppose that is rational and cuspidal, meaning that it is rational and all its singularities are cusps. A curve is free or nearly free according to the corresponding properties of its Jacobian syzygies. Rational cuspidal curve conjecture. The curve is either free or nearly free. This conjecture proposes a strong restriction on the syzygies of rational cuspidal plane curves; the source presents it as a motivating conjecture, but the supplied text does not state whether it has been resolved.
source: Alexandru Dimca and Gabriel Sticlaru, “Plane curves with three syzygies, minimal Tjurina curves curves, and nearly cuspidal curves”, arXiv:1810.11766 (2019).
Sources & referencesView supporting material
Primary source
Alexandru Dimca and Gabriel Sticlaru, “Nearly free divisors and rational cuspidal curves”, arXiv:1505.00666 (2015).
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