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Rationality of moduli spaces of plane curves of small degree. (English) Zbl 1184.14025

Let \(C(d) = {\mathbb P}(\mathrm{Sym}^d({\mathbb C}^3)^*/\mathrm{SL}_3({\mathbb C})\) be the moduli space of plane curves of degree \(d\). The rationality of \(C(d)\) for big values of \(d\) is known:
(1) for \(d \equiv 0\) (mod. 3) and \(d \geq 210\) – by P. I. Katsylo [Math. USSR, Sb. 64, No. 2, 375–381 (1989); translation from Mat. Sb., Nov. Ser. 136(178), No. 3(7), 377–383 (1988; Zbl 0679.14028)];
(2) for \(d \equiv 1\) (mod. 3) and \(d \geq 37\), and \(d \equiv 2\) (mod. 3) and \(d \geq 65\) – by C. Böhning and H.-C. Graf von Bothmer [Invent. Math. 179, No. 1, 159–173 (2010; Zbl 1188.14009)] or [arxiv:0804.1503];
(3) for \(d \equiv 1\) (mod. 4) – by N. I. Schepherd-Barron [Algebraic geometry, Proc. Summer. Res. Inst., Brunswick/Maine 1985, Part 1, Proc. Symp. Pure Math. 46, No. 1, 165–171 (1987; Zbl 0669.14015)]. Thus the question for rationality of \(C(d)\) remains open for the small values of \(d\) not covered by (1)-(2)-(3). Here the authors show the rationality of \(C(d)\) for most of these small values: The moduli space \(C(d)\) is rational except possibly if \(d = 6,7,8,11,12,14,15,16,18,20,23,24,26,32\) and \(48\), which is the main result of the paper.

MSC:

14E08 Rationality questions in algebraic geometry
14M20 Rational and unirational varieties
14L24 Geometric invariant theory
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