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Plane curves with aligned conductors. (English) Zbl 1058.14054
Summary: An invariant toward a classification of curves \(X\) with a birationally very ample and special invertible sheaf \(L\) is proposed, and an extremal pair \((X, L)\) by means of the invariant is a smooth plane curve \(X\) with \(O_X(1)\) corresponding line sections and vice versa. For pairs \((X, L)\) next to extremal ones, all but one exception is characterized by the image \(\phi_L(X)\) to be a plane singular curve with aligned conductors. The heart of the paper is to study such a singular plane curve with its normalization involving projective geometry and the theory of adjoint curves.
MSC:
14H50 Plane and space curves
14H45 Special algebraic curves and curves of low genus
14H51 Special divisors on curves (gonality, Brill-Noether theory)
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