ABSTRACT

In this chapter we describe A5-orbits of lengths 20 and 30 on V5. Together with Theorem 7.3.5, this gives a complete classification of A5-orbits on V5. We show that all A5-orbits of length 20 except for Σ

′ 20 are contained in a

certain A5-irreducible curve G10 that is a union of 10 conics, and all A5orbits of length 30 are contained either in the curve L15 or in a certain A5-irreducible curve T15 that is a union of 15 twisted cubics. We also study basic properties of the curves G10 and T15. It should be pointed out that many of these results can be proved by a direct computation (see Lemmas 5.5.13 and 5.5.14). Nevertheless, we prefer to use a slightly longer and more straightforward approach, due to our general policy to avoid using explicit equations of V5 and related computations.