# On the construction of Weakly Ulrich bundles

## TL;DR

This paper constructs large-rank weakly Ulrich bundles, called almost Ulrich bundles, on various algebraic surfaces and threefolds in positive characteristic, expanding the known classes of such bundles.

## Contribution

It introduces a method to construct intrinsic weakly Ulrich bundles, including ACM and almost Ulrich bundles, on a broad class of surfaces and threefolds in characteristic p>0.

## Key findings

- Constructed weakly Ulrich bundles on smooth surfaces in P^3 in characteristic p>0.
- Extended constructions to Frobenius split varieties of dimension up to three.
- Identified classes of varieties, including hypersurfaces and Calabi-Yau varieties, admitting these bundles.

## Abstract

I provide a construction of intrinsic weakly Ulrich bundles of large rank on any smooth complete surface in ${\bf P}^3$ over fields of characteristic $p>0$ and also for some classes of surfaces of general type in ${\bf P}^n$. 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 ${\bf P}^4$ of degree at most four, (2) Frobenius split, smooth quintics in ${\bf P}^4$ (3) Frobenius split Calabi-Yau varieties of dimension at most three (4) Frobenius split (i.e ordinary) abelian varieties of dimension at most three.