Compactifications of contractible affine 3-folds into smooth Fano 3-folds with B2=2

Abstract

After the contributions of Furushima, Nakayama, Peternell and Schneider, in 1993, Furushima [Fur93] finally succeeded in the classification of the compactifications of the affine 3-space Open image in new window into smooth Fano 3-folds with B2=1. In this paper, we consider the compactifications of the contractible affine 3-folds X (not necessarily X=Open image in new window) into smooth Fano 3-folds V with B2=2. Consequently, we classify all such compactifications X↪(V,D1∪D2) in the case where KV+D1+D2 is not nef. Furthermore, we see that infinitely many mutually non-isomorphic exotic Open image in new window's can be compactified into Fano 3-folds with B2=2. This phenomenon never occurs when B2=1.