In this work we characterize Ulrich bundles of any rank on polarized rational ruled surfaces over \({\mathbb {P}^1}\). We show that every Ulrich bundle admits a resolution in terms of line bundles. Conversely, given an injective map between suitable totally decomposed vector bundles, we show that its cokernel is Ulrich if it satisfies a vanishing in cohomology. As a consequence we obtain, once we fix a polarization, the existence of Ulrich bundles for any admissible rank and first Chern class. Moreover we show the existence of stable Ulrich bundles for certain pairs \(({\text {rk}}(E),c_1(E))\) and with respect to a family of polarizations. Finally we construct examples of indecomposable Ulrich bundles for several different polarizations and ranks.

在这项工作中，我们在\（{\ mathbb {P} ^ 1} \）上的极化有理直纹曲面上表征了任意等级的Ulrich束。我们表明，每个Ulrich束都接受线束的分辨率。相反，给定合适的完全分解向量束之间的内射图，我们证明，如果它的同位性消失，则其内核为Ulrich。结果，一旦确定了极化，我们就可以得出任何允许的秩和第一类Chern类的Ulrich束的存在。此外，我们证明了对于某些对\（（{{text {rk}}（E），c_1（E））\）以及关于极化族的稳定Ulrich束的存在。最后，我们为几个不同的极化和秩构造了不可分解的Ulrich束的例子。