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Projective construction creates algebraic cubics from line pencils, suggesting new geometric insights.
We give a projective construction for general non-singular algebraic plane cubics using a (2,2) correspondence between two line pencils. When the pencils are equipped with homographic involutions, the residual locus of their intersections---obtained by removing the baseline---is a cubic curve passing through the centers of both pencils. We also describe the tangency properties of the fixed lines of these involutions relative to the total quartic locus.
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Hussein Khayou (2026) studied this question.
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