On the extendability of elliptic surfaces of rank two and higher
A natural approach to study projective algebraic variety is the so called Apollonius method. That is consider an hyperplane section and then reconstruct the geometry of the ambient variety via the one of its section. It is therefore quite natural to understand which varieties can be embedded as hyperplane sections of other varieties. Besides many classical results, a fundamental work of \textit{A. J. Sommese} [Math. Ann. 221, 55--72 (1976; Zbl 0306.14006)] gave many criteria for extendability and non extendability of algebraic varieties. This area is called adjunction theory [\textit{M. C. Beltrametti, J. J. Sommese}, The adjunction theory of complex projective varieties. De Gruyter Expositions in Mathematics. 16. (Berlin): de Gruyter. (1995; Zbl 0845.14003)], and attracted many attention in the last decades, also for its relations with Minimal Model Program. The paper under review considers this problem for smooth surfaces that admits an elliptic fibrations. The main result is a list of the possible embeddings of these surfaces, of Picard rank 2, as hyperplane sections of mildly singular 3-folds. This is done via a subtle study of the geometry of the surface in relation with its possible three dimensional extensions. The result is then improved with stronger hypothesis on the surface and many interesting examples are given.
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