Rationality of moduli spaces of plane curves

From CRCG-Wiki

(Joint work with Christian Boehning)

It is a classical question, which can be traced back to works of Hilbert and Emmy Noether, whether the orbit spaces P/G are rational where P is a pro jective space and G is a reductive algebraic group acting linearly in P. If G is not assumed connected, in fact for G a finite solvable group, D. Saltman has shown that the answer to this question is negative in general (Emmy Noether had apparently conjectured that the quotient should be rational in this case). No counterexamples are known for connected complex reductive groups G.

A geometrically relevant case is the moduli space of plane curves of degree d. By works of Bogomolov it is known that these spaces are stably rational. Unfortunately there exist examples of stably rational that are not rational, so that this fact alone can not imply the rationality of these moduli spaces.

Known results:

• d ≤ 3 (well known),
• d = 4 (Katsylo, very difficult),
• d ≡ 1 (mod 4) (Shepherd-Barron),
• d ≡ 1 (mod 9) and d ≥19 (Shepherd-Barron);
• d ≡ 0 (mod 3) and d ≥ 1821 (Katsylo, Bogomolov)

We are able to prove rationality in the following cases:

• d ≡ 1 (mod 3) and d ≥ 37
• d ≡ 2 (mod 3) and d ≥ 65

For this we use a combination of classical methods and computer algebra:

• The symbolic method of Aronhold and Clebsch (1903)
• Writing polynomials as sums of powers (also classical)
• Covariants (in a generalization of an idea of Shepherd-Barron)
• reduction modul p
• calculating the rank of finitely many small matrices using Macaualay2