Weierstrass normal forms and invariants of elliptic surfaces

Abstract

Let π : S → B \pi :S \to B be an elliptic surface with a section σ : B → S \sigma :B \to S . Let L − 1 → B {L^{ - 1}} \to B be the normal bundle of σ ( B ) \sigma (B) in S, and let W = P ( L ⊗ 2 ⊕ L ⊗ 3 ⊕ 1 ) W = P({L^{ \otimes 2}} \oplus {L^{ \otimes 3}} \oplus 1) be a P 2 {{\mathbf {P}}^2} -bundle over B. Let S ∗ {S^\ast } be the surface obtained from S by contracting those components of fibres of S which do not intersect σ ( B ) \sigma (B) . Then S ∗ {S^\ast } may be imbedded in W and defined by a “Weierstrass equation": \[ y 2 z = x 3 − g 2 x z 2 − g 3 z 3 {y^2}z = {x^3} - {g_2}x{z^2} - {g_3}{z^3} \] where g 2 ∈ H 0 ( B , O ( L ⊗ 4 ) ) {g_2} \in {H^0}(B,\mathcal {O}({L^{ \otimes 4}})) and g 3 ∈ H 0 ( B , O ( L ⊗ 6 ) ) {g_3} \in {H^0}(B,\mathcal {O}({L^{ \otimes 6}})) . The only singularities (if any) of S ∗ {S^\ast } are rational double points. The triples ( L , g 2 , g 3 ) (L,{g_2},{g_3}) form a set of invariants for elliptic surfaces with sections, and a complete set of invariants is given by { ( L , g 2 , g 3 ) } / G \{ (L,{g_2},{g_3})\} /G where G ≅ C ∗ × Aut ( B ) G \cong {{\mathbf {C}}^\ast } \times {\operatorname {Aut}}\;(B) .