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On the projective invariants of quartic plane curves. (English) Zbl 0668.14006
Let us consider the ternary quartic forms, i.e., the homogeneous polynomials of degree \( 4\) in three variables, with complex coefficients. The set of these forms is a complex vector space V, of dimension 15. Let \({\mathbb{C}}[V]\) be the algebra of complex polynomial functions on V. Let \({\mathbb{C}}[V]\) be the algebra of complex polynomial functions on V. Let \({\mathbb{C}}[V]_ n\) be the set of homogeneous elements of degree n in \({\mathbb{C}}[V]\). Then \({\mathbb{C}}[V]=\oplus_{n\geq 0}{\mathbb{C}}[V]_ n\) is a graded algebra.
The group SL(3,\({\mathbb{C}})\) operates in a natural way in V, hence in \({\mathbb{C}}[V]\), leaving stable every \({\mathbb{C}}[V]_ n\). Let \({\mathcal A}\) be the set of elements in \({\mathbb{C}}[V]\) which are invariant under SL(3,\({\mathbb{C}})\). Then \({\mathcal A}\) is a graded subalgebra of \({\mathbb{C}}[V]\), the algebra of projective invariants of quartic plane curves. Let \({\mathcal A}_ n={\mathcal A}\cap {\mathbb{C}}[V]_ n\). T. Shioda stated very precise conjectures concerning the structure of \({\mathcal A}\) [Am. J. Math. 89, 1022-1046 (1967; Zbl 0188.533); p. 1046]. We will prove the first of these conjectures: \({\mathcal A}\) admits a homogeneous system of parameters, of degrees 3, 6, 9, 12, 15, 18, 27. (Our result will be more precise, since we will exhibit an explicit system of parameters with these degrees.)

14L24 Geometric invariant theory
14L30 Group actions on varieties or schemes (quotients)
11E76 Forms of degree higher than two
Full Text: DOI
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