# zbMATH — the first resource for mathematics

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.)

##### MSC:
 14L24 Geometric invariant theory 14L30 Group actions on varieties or schemes (quotients) 11E76 Forms of degree higher than two
Full Text:
##### References:
 [1] Elliott, E.B, An introduction to the algebra of quantics, (1913), Oxford Univ. Press London/New York · JFM 44.0155.05 [2] Gordan, P, Über Büschel von kegelschnitten, Math. ann., 19, 529-552, (1882) · JFM 14.0080.01 [3] Hilbert, D, Gesammelte abhandlungen, (1965), Chelsea New York · JFM 59.0037.06 [4] Popov, V.L, Constructive invariant theory, Astérisque, 87-88, 303-334, (1981) · Zbl 0491.14004 [5] Salmon, G, Higher plane curves, (1879), reprinted by, Chelsea, New York [6] Shioda, T, On the graded ring of invariants of binary octavics, Amer. J. math., 89, 1022-1046, (1967) · Zbl 0188.53304
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.