Abstract
AbstractWe classify rank two vector bundles on a given del Pezzo threefold of degree four whose projectivizations are weak Fano into seven cases. We also give an example for each of these seven cases.
Highlights
Since smooth Fano 3-folds were classified, many researchers have treated the classification of Fano 4-fold having projective bundle structures
The classification of rank 2 Fano bundles over smooth Fano 3-folds has been addressed by many researchers (e.g. [22, 27, 28, 37])
Takeuchi developed his 2-ray game method by considering Fano 3-folds and weak Fano 3-folds of Picard rank 2 [38]. By this 2-ray game method, he successfully gave a concise proof of the existence of a line on a Mukai 3-fold, which is a new perspective on the classification of Fano 3-folds
Summary
The motivation of this study comes from the classification of Fano manifolds. Since smooth Fano 3-folds were classified (see [18] and references therein), many researchers have treated the classification of Fano 4-fold having projective bundle structures. Takeuchi developed his 2-ray game method by considering Fano 3-folds and weak Fano 3-folds of Picard rank 2 [38]. Since the establishment of the 2-ray game method, classifying weak Fano 3-folds of Picard rank 2 has been considered to be significant and treated by many researchers In view of these previous researches, classifying weak Fano 4-folds with Picard rank 2 would be important to investigate Fano 4-folds Our approach to this problem is to consider weak Fano 4-folds with P1-bundle structures, as Szurek–Wisniewski did [37]
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