Pencils of binary quartics (Q1126555)

In 1983, \textit{N. D. Beklemishev} [Math. USSR, Izv. 23, 423-429 (1984); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 47, No. 6, 1155-1161 (1983; Zbl 0582.14019)] showed that the ring of invariants of a pencil of \(d\)-ics in \(r\) variables is a complete intersection only in the cases \(d\leq 3\), \(r\leq 4\) and \(d\leq 6\), \(r=2\). In the paper under review, the author considers pencils of quartics in two variables, giving a complete description of the ring of invariants. It is interesting to note that this problem had already been studied in 1866 by G. Salmon, who however overlooked one of the invariants. The proof follows the argument given by \textit{P. E. Newstead} for pencils of cubics [Math. Proc. Camb. Philos. Soc. 89, 201-209 (1981; Zbl 0458.14006)]. The computations are rather involved and have been performed also using the computer algebra program Maple. -- Then the author describes non-semistable and non-stable pencils and gives a stratification of the moduli space.