Complex Algebraic Surfaces 2026, 47 - Ruled Surfaces

A surface S is ruled when birationally equivalent to C × ℙ¹ where C is a smooth curve. If C = ℙ¹ this extends to be rational (III.1). If C is smooth, with a geometrically ruled surface S and a smooth morphism p: S → C whose fibres are isomorphic to ℙ¹ (III.3). A surface S and smooth curve C with morphism p: S → C, and an x ∈ C : p is smooth over x &land; the fibre p^-1(x) is isomorphic to ℙ¹, then there is a Zariski open subset U of C containing x and an isomorphism from p^-1(U) to U × ℙ¹ that creates a commutative diagram. S is ruled (III.4, Nother-Enriques). If D is an effective divisor and C an irreducible curve with C²≥ 0 ⇒ D.C ≥ 0 (III.5). Every geometrically ruled surface over C is C-isomorphic to ℙ_C(E) for a rank-2 vector bundle over C. ℙ_C(E), ℙ_C(E') are C-isomorphic iff ∃ line bundle L on C s.t. E' ≡ E ⊗ L (III.7). A smooth irrational curve C has minimal models of C × ℙ¹ are geometrically ruled over C, which comprise ℙ_C(E) (III.10). There is an exact sequence 0 → L → E → M → 0 for L, M ∈ Pic C. If h⁰(E) ≥ 1, L = O_C(D) with D ≥ 0; if h⁰(E) ≥ 2 and deg(E) > 0, then D > 0 (III.11). χ(E) = deg(E) + 2 - 2g(C) (III.12, Riemann-Roch for Rank 2 vector bundles). The extension of M by L as in III.11 implies that 0 → L ⊗ M^-1 → E ⊗ M^-1 → O_C → 0 iff it there is a section of H⁰(C, E ⊗ M^-1) mapping to 1 ∈ H⁰(C, C_C) (III.14). Given S = ℙ_C(E), with structural morphism p: S → C, the bundle p^*E on S has a line bundle N as a sub-bundle. Above some p ∈ S, it corresponds to a line D ⊂ E_p(s), N_s = D. O_S(1) is the tautological bundle of S, defined as 0 → N → p^*E →^u O_S(1) → 0 (III.17). A variety T with morphism f: T → C associates a line bundle L = g^*O_S(1) to C-morphisms g: T → ℙ_C(E). Conversely, given a line bundle L on T and a surjective morphism v: f^*E → L, defining g by associating lines Ker(v_t) to a point t in T. The inverse construction also works.

Given a surface S, define a number of algebro-geometric invariant integers.

• q(S) = h¹(S, O_S)
• p_g(S) = h²(S, O_S) = h⁰(S, O_S(K)) (as per Serre)
• P_n(S) = h⁰(S, O_S(nK)) ∀ n ≥ 1

P_n are the plurigenera of S, p_g = P₁ is the geometric genus, q is the irregularity of S. They are birational invariants (III.20). By Noether's formula, q(S) = h⁰(S, Ω¹_S) = 1/2 b₁(S) (III.19). For a ruled surface S over C, q(S) = g(C); p_g(S) = 0; P_n(S) = 0 ∀ n ≥ 2. If S is geometrically ruled, K_S² = 8(1 - g(C)); b₂(S) = 2 (III.21). Given varieties X, Y; p, q projections of X × Y on X, Y, then if F is a vector bundle on X and G on Y, the emerging canonical homomorphism is isomorphic. If X, Y are smooth, Ω¹_{X × Y} ≡ p^*Ω¹_X ⊕ q^*Ω¹_Y (III.22).