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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)
##### Keywords:
birationally very ample; theory of adjoint curves