On the construction of weakly Ulrich bundles

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 P³ over fields of characteristic p>0 and also for some classes of surfaces of general type in Pⁿ. 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 P⁴ of degree at most four, (2) more generally, Frobenius split Fano varieties of dimension at most three, (3) Frobenius split, smooth quintics in P⁴ (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.