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Quotient of the variety of infinitely near points of order 9 under the action of \(\text{PGL}_ 3\). (Quotient de la variété des points infiniment voisins d’ordre 9 sous l’action de \(\text{PGL}_ 3\).) (French) Zbl 0870.14017
Let \(S_n\) denote the variety of infinitely near points of order \(n\) to points of the projective plane. The author studies the natural action of \(G := \text{PGL}(3, {\mathbb{C}})\) on \(S_n\), using the differential invariants introduced by G.H. Halphen. It is shown that the field of invariant rational functions of \(S_n\) is purely transcendental over \({\mathbb{C}}\). This is applied to obtain rationality for moduli spaces of pointed plane curves of a given degree. For the cases \(n=8,9\) an explicit construction of open \(G\)-stable subsets of \(S_n\) that admit a quotient by the action of \(G\) is presented.

MSC:
14H10 Families, moduli of curves (algebraic)
14L30 Group actions on varieties or schemes (quotients)
14G05 Rational points
14B10 Infinitesimal methods in algebraic geometry
14M20 Rational and unirational varieties
14L24 Geometric invariant theory
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References:
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