Abstract
Let us now take up the study of the solution set E⊆ℂℙ2 of the equation
where F is a homogeneous polynomial of degree 3. Again we will assume that the partial derivatives
do not all vanish simultaneously. Unlike the degree-2 case, every cubic polynomial with real coefficients has at least one real root, so if the coefficients of F are real, then the solution set to (2.1) in ℝℙ2will always be a smooth curve. The question whether there exists a solution in ℚℙ2if the coefficients of (2.1) are rational is extremely difficult, and there is as yet no known procedure for deciding in general.
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© 1980 Plenum Press, New York
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Clemens, C.H. (1980). Cubics. In: A Scrapbook of Complex Curve Theory. The University Series in Mathematics. Springer, Boston, MA. https://doi.org/10.1007/978-1-4684-7000-0_2
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DOI: https://doi.org/10.1007/978-1-4684-7000-0_2
Publisher Name: Springer, Boston, MA
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