zbMATH — the first resource for mathematics

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.
14H50 Plane and space curves
14H45 Special algebraic curves and curves of low genus
14H51 Special divisors on curves (gonality, Brill-Noether theory)