The moduli space of binary quintics (Q1746646)

The \(\mathrm{SL}_2\)-invariant ring of binary quintics over \(\mathbb{C}\) is generated by four invariants \(I_4\), \(I_8\), \(I_{12}\), and \(I_{18}\). Their moduli space \(\mathcal M\) (with quintics with cubic factors removed) admits coordinates \((I_4, I_8, I_{12})\). It contains the curve \(\Delta\) of quintics with repeated roots and the curve \(\Gamma_5\) of quintics with nontrivial symmetry. The authors describe the topology of real points of \(\mathcal M\) with prescribed numbers of real roots and consider in more detail five exceptional curves in \(\mathcal M\) that arose in [\textit{C. T. C. Wall}, Math. Proc. Camb. Philos. Soc. 119, No. 2, 257--277 (1996; Zbl 0868.14015)]: There are five special quintics \(H_1\), \(H_2\), \dots, \(H_5\) in \(\mathbb{P}^2\) with one-parametric symmetry group. The intersection points with the corresponding orbits of lines give curves \(\Gamma_1\), \(\Gamma_2\), \dots, \(\Gamma_5\) in \(\mathcal M\). Their equations, intersections with \(\Delta\), mutual intersections, and singularities are computed. In the concluding section, the theory is adapted to characteristic two.