ABSTRACT
In this chapter we classify A5-irreducible curves of low degree contained in V5 (cf. Lemma 8.1.14 and Proposition 10.2.5) and study A5-invariant anticanonical divisors passing through them.
Recall from Lemma 8.1.16(i) that the surface S ⊂ V5 contains a unique A5-irreducible curve C16 of degree 16. This curve is smooth and rational by Lemma 8.1.16(ii). The curve C16 is not the only A5-irreducible curve of degree 16 contained in V5. In this section we construct an A5-irreducible curve C ′16 of degree 16 in V5 different from C16, and prove that V5 does not contain other A5-irreducible curves of degree 16. We will use notation of §12.5. Lemma 13.1.1. Let C ⊂ V5 be an irreducible A5-invariant curve of degree 16 that is not contained in the surfaceS . Then the following assertions hold:
(i) one has C ∩S = Σ′12 ∪ Σ′20; (ii) the curve C is disjoint from the A5-orbits Σ12 and Σ20 that are con-
tained in S ;
(iii) one has C ∩ C ′ = Σ′12; (iv) the curves C and C ′ intersect transversally at the points of Σ′12.
