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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Weierstrass normal forms and invariants of elliptic surfaces

Author: Arnold Kas
Journal: Trans. Amer. Math. Soc. 225 (1977), 259-266
MSC: Primary 14J25; Secondary 14K05
MathSciNet review: 0422285
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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":

$\displaystyle {y^2}z = {x^3} - {g_2}x{z^2} - {g_3}{z^3}$

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)$.

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Keywords: Elliptic surfaces, Weierstrass equation, rational double points, minimal resolutions
Article copyright: © Copyright 1977 American Mathematical Society