In this chapter we present the classification theory by Δ-genus. The main technique is the hyperplane section method using induction on the dimension.

Characterizations of projective spaces

First of all, as the most typical example of the Apollonius method, we recall the proof of the following.

(1.1) Theorem (cf. [Gor], [KobO]). Let (V, L) be a polarized variety such that n = dim V, Ln = 1 and h0 (V, L) ≥ n + 1. Assume that V has only Cohen Macaulay singularities. Then (V, L) ≃ (ℙn, o(1)).

Proof. We use the induction on n. The case n = 1 is easy, so we consider the case n ≥ 2. Take a member D of │L│. Then Ln−1D = Ln = 1. If D = D1 + D2 for non-zero effective Weil divisors Di, then Ln−1Di > O and Ln−1D ≥ 2 since L is ample. Therefore D is irreducible and reduced as a Weil divisor. On the other hand, D has a natural structure as a subscheme of V such that Coker(δ) ≃ OD for the homomorphism δ: OV[−L] → OV defining D. By the above observation Supp(D) is irreducible and the scheme D is reduced at its generic point. V has only Cohen Macaulay singularities, hence so does D. Therefore D has no embedded component and is reduced everywhere, so D is a variety.