Weierstrass normal forms and invariants of elliptic surfaces

Abstract:

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