The goal of this paper is to study the existence of Ulrich bundles on the imprimitive Fano 3-folds obtained blowing-up one of the following varieties along a smooth irreducible curve: the projective space \(\mathbb{P}^3\), the smooth quadric \(Q\subset \mathbb{P}^4\), a smooth cubic \(V_3\subset \mathbb{P}^4\) or the complete intersection of two quadrics \(V_4 \subset \mathbb{P}^5\).
Recall that a vector bundle \(\mathcal{F}\) on a \(n\)-dimensional projective variety \(X \subset \mathbb{P}^N\) is called Ulrich if \[ H^i(X,\mathcal{F}(-t))=0 \mbox{ for any } i \geq0 \mbox{ and } 1 \leq t \leq n. \] In [D. Eisenbud and F.-O. Schreyer, J. Am. Math. Soc. 16, No. 3, 537–575, appendix by J. Weyman 576–579 (2003; Zbl 1069.14019)], it is conjectured that every variety admits an Ulrich bundle and, once the existence is proven, it is pointed out, as an interesting problem, to determine its minimal rank.
The main result of this paper states that, among the varieties considered, the only ones admitting a rank 1 Ulrich bundle are given by blowing up \(\mathbb{P}^3\) along a curve of genus 3 and degree 6 (see Section 2). Focusing on these varieties, polarized by \(H\), the author shows that \(X\) admits two classes of rank 1 Ulrich bundles (see Section 3). Finally, he proves the existence of stable rank 2 Ulrich bundles with \(c_1 = 3H\) (see Section 4).