Maximal-rank conjecture for focal maps of equiclassical plane curves
Let be an irreducible component of the real equiclassical family of curves in of degree , genus , and class , and let be a general smooth point of . Assume that the usual equiclassical tangent-space description and adjoint sequences hold at , and that the focal adjoint bundle satisfies
this holds, for instance, in the nodal-cuspidal case. Maximal-rank conjecture. Then
Consequently, the focal map has maximal rank at :
This predicts that the focal deformation space is governed by the focal adjoint bundle and that the focal map imposes the expected number of independent conditions. The supplied text does not state whether this claim has been proved or disproved.
References
Primary source
Ragni Piene and Boris Shapiro, “Confocal families of plane algebraic curves”, arXiv:2604.25293 (2026).
Additional references
8 papers in this index state this conjecture (2002–2026). The statement above is taken from the most recent of them; the others are arXiv:2302.04513, arXiv:1910.08843, arXiv:1503.06762, arXiv:1108.4714, arXiv:0906.3870, arXiv:math/0212146, arXiv:math/0208120.
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