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

DOI:
https://doi.org/10.1090/S0002-9947-1977-0422285-X

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": \[ {y^2}z = {x^3} - {g_2}x{z^2} - {g_3}{z^3}\] 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 https://doi.org/10.1007/BF01361440 - F. Hirzebruch, W. D. Neumann, and S. S. Koh,
*Differentiable manifolds and quadratic forms*, Marcel Dekker, Inc., New York, 1971. Appendix II by W. Scharlau; Lecture Notes in Pure and Applied Mathematics, Vol. 4. MR**0341499**
K. Kodaira,

*On compact analytic surfaces*. II, Ann. of Math. (2)

**77**(1963), 563-626. MR

**32**#1730.

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Keywords:
Elliptic surfaces,
Weierstrass equation,
rational double points,
minimal resolutions

Article copyright:
© Copyright 1977
American Mathematical Society