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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Weierstrass normal forms and invariants of elliptic surfaces
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by Arnold Kas PDF
Trans. Amer. Math. Soc. 225 (1977), 259-266 Request permission


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 $G \cong {{\mathbf {C}}^\ast } \times {\operatorname {Aut}}\;(B)$.
  • Egbert Brieskorn, Über die Auflösung gewisser Singularitäten von holomorphen Abbildungen, Math. Ann. 166 (1966), 76–102 (German). MR 206973, DOI 10.1007/BF01361440
  • F. Hirzebruch, W. D. Neumann, and S. S. Koh, Differentiable manifolds and quadratic forms, Lecture Notes in Pure and Applied Mathematics, Vol. 4, Marcel Dekker, Inc., New York, 1971. Appendix II by W. Scharlau. MR 0341499
  • K. Kodaira, On compact analytic surfaces. II, Ann. of Math. (2) 77 (1963), 563-626. MR 32 #1730.
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Additional Information
  • © Copyright 1977 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 225 (1977), 259-266
  • MSC: Primary 14J25; Secondary 14K05
  • DOI:
  • MathSciNet review: 0422285