## 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

### Keywords:

rationality; moduli spaces; plane curves; group quotients

### Citations:

Zbl 0679.14028; Zbl 0669.14015; Zbl 1188.14009

### Software:

ClebschGordan; smallDegree; Macaulay2; rationality; FFLAS-FFPACK
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