Classification of rank two weak Fano bundles on del Pezzo threefolds of degree four

Abstract

We 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.

Rank two weak Fano bundles on del Pezzo threefolds of Picard rank one

“…Classifications on del Pezzo 3-folds of degree ≤ 2. By Theorem 1.1 and our previous studies [13,8], we obtain the classification result on del Pezzo 3-folds X of degree 3 ≤ deg X ≤ 5. To complete our classification, we show there exist no nontrivial weak Fano bundles when the case deg X ≤ 2, which is namely the following theorem.…”

“…Briefly speaking, we showed that every indecomposable weak Fano bundle is either a Fano bundle or a vector bundle E with c 1 (E) = 0 such that E(1) is globally generated. By the same approach as in [8], we will see that this phenomenon still holds when deg X = 5 (see Section 5.3).…”

“…It was known that a del Pezzo 3-fold is of Picard rank 1 if and only if the degree of X, say deg X, is less than or equal to 5 [14,6]. In our previous articles [13,8], we classify rank 2 weak Fano bundles on a del Pezzo 3-fold X of deg X ∈ {3, 4}. In this sequel, we will classify them when the case deg X ∈ {1, 2, 5}.…”

“…Known Results. In our previous work [8], we treat this kind of problem when deg X = 4. Briefly speaking, we showed that every indecomposable weak Fano bundle is either a Fano bundle or a vector bundle E with c 1 (E) = 0 such that E(1) is globally generated.…”

“…While instanton bundles E with c 2 (E) ≤ 3 were studied by Sanna [26,27] in detail, those with c 2 (E) = 4 have not been studied yet. Thus, we need a different approach from our previous studies [13,8] to investigate such weak Fano bundles.…”