Maximal-rank conjecture for focal maps of equiclassical plane curves

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Let WW be an irreducible component of the real equiclassical family of curves in (P2)∨(\mathbb{P}^2)^\vee of degree cc, genus gg, and class dd, and let GG be a general smooth point of WW. Assume that the usual equiclassical tangent-space description and adjoint sequences hold at D={G=0}D=\{G=0\}, and that the focal adjoint bundle BD\mathcal{B}_D satisfies

deg⁡BD=d−c;\deg \mathcal{B}_D=d-c;

this holds, for instance, in the nodal-cuspidal case. Maximal-rank conjecture. Then

h0(D~,BD)=max⁡(0,d−g−c+1).h^0(\widetilde D,\mathcal{B}_D)=\max(0,d-g-c+1).

Consequently, the focal map has maximal rank at GG:

rank⁡(dΦG)=min⁡(2c,c+d−g+1).\operatorname{rank}(d\Phi_G)=\min(2c,c+d-g+1).

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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