A well-known conjecture of Eisenbud, Schreyer and Weyman suggests that any projective variety carries an Ulrich (and hence also weakly Ulrich) bundle. This is known only in a handful of cases. In this paper I provide weakly Ulrich bundles on a class of surfaces and threefolds. I construct a weakly Ulrich bundle of large rank on any smooth complete surface in ${\mathbb{P}}^3$ over fields of characteristic $p>0$ and also for some classes of surfaces of general type in ${\mathbb{P}}^n$. I also construct intrinsic weakly Ulrich bundles on any Frobenius split variety of dimension at most three. The bundles constructed here are in fact ACM and weakly Ulrich bundles and so I call them almost Ulrich bundles. Frobenius split varieties in dimension three include as special cases: (1) smooth hypersurfaces in ${\mathbb{P}}^4$ of degree at most four, (2) more generally, Frobenius split Fano varieties of dimension at most three, (3) Frobenius split, smooth quintics in ${\mathbb{P}}^4$ (4) more generally, Frobenius split Calabi-Yau varieties of dimension at most three (5) Frobenius split (i.e. ordinary) abelian varieties of dimension at most three. These results also imply that Chow form of these varieties is the support of a single intrinsic determinantal equation.

Eisenbud，Schreyer和Weyman的一个著名猜想表明，任何射影变种都带有Ulrich（因此也是弱Ulrich）束。这仅在少数情况下是已知的。在本文中，我提供了一类曲面和三重曲面上的弱Ulrich束。我在的任何光滑完整表面上构造了一个弱弱的大阶Ulrich束${\mathbb{P}}^3$ 在特征领域 $p>0$ 对于某些类型的一般类型的曲面 ${\mathbb{P}}^{ñ}$。我还会在最多三个维的任何Frobenius分裂维上构造固有的弱Ulrich束。这里构造的束实际上是ACM和弱Ulrich束，因此我将它们称为Ulrich束。第3维的Frobenius分裂变体包括以下特殊情况：（1）光滑的超曲面${\mathbb{P}}^4$ 的度数最多为四个，（2）更普遍的是Frobenius分裂维诺变种的尺寸最多为三个，（3）Frobenius分裂，光滑的五分之一 ${\mathbb{P}}^4$（4）更一般地，Frobenius分裂维数最多的卡拉比尤丘变种（5）Frobenius分裂（即普通）维数最多的三个阿贝尔变种。这些结果还暗示这些品种的Chow形式是单个内在行列式方程的支持。