On compactifications of affine homology 3-cells into quadric fibrations

Abstract

In this paper we deal with compactifications of affine homology 3-cells into quadric fibrations such that the boundary divisors contain fibers. We show that all such affine homology 3-cells are isomorphic to the affine 3-space A 3 . Moreover, we show that all such compactifications can be connected by explicit elementary links preserving A 3 to the projective 3-space P 3 . Contents 1. Introduction. 1 2. Elementary links 6 2.1. Elementary links between blow-ups 7 2.2. Elementary links between P 2 -bundles 7 2.3. Elementary links from quadric fibrations to P 2 -bundles 8 2.4. Elementary links between quadric fibrations 10 3. Topological invariants of the ambient space 11 4. Proof of Theorem 1.3. 11 5. Examples 14 6. Compactifications of affine homology 3-cells compatible with P 2 -bundles 16 7. Proof of Theorem 1.8 18 7.1. Singularities of D h and D f | D h 18 7.2. The case of singular D f | D h 21 7.3. The case of smooth D f | D h 22 Acknowledgement 23 References 24

$\mathbb{G}_a^3$-structures on del Pezzo fibrations

“…The main step to prove this implication is Proposition 3.4, that is, the exclusion of the case when the degrees of del Pezzo fibrations are eight. For this, we use the results of [Nag19]. Finally, we prove the opposite implication in §4.…”