Chapter I - Δ-genus and Apollonius Method

Summary

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], [K_obO]). 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 D_i, then Lⁿ⁻¹D_i > 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) ≃ O_D for the homomorphism δ: O_V[−L] → O_V 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.