Characterization of Ulrich bundles on Hirzebruch surfaces

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 $(\textrm{rk}(E),c_1(E))$ and with respect to a family of polarizations. Finally we construct examples of indecomposable Ulrich bundles for several different polarizations and ranks.Comment: 23 pages. Incorporated Section 5 in the other sections. Added Section 6 on existence and moduli space of Ulrich bundles. Final version in Revista Matematica Complutens

Ulrich bundles on ruled surfaces

In this short note, we study the existence problem for Ulrich bundles on ruled surfaces, focusing our attention on the smallest possible rank. We show that existence of Ulrich line bundles occurs if and only if the coefficient $\alpha$ of the minimal section in the numerical class of the polarization equals one. For other polarizations, we prove the existence of rank two Ulrich bundles

Ulrich sheaves and higher-rank Brill-Noether theory

An Ulrich sheaf on an embedded projective variety is a normalized arithmetically Cohen-Macaulay sheaf with the maximum possible number of independent sections. Ulrich sheaves are important in the theory of Chow forms, Boij-Soderberg theory, generalized Clifford algebras, and for an approach to Lech's conjecture in commutative algebra. In this note, we give a reduction of the construction of Ulrich sheaves on a projective variety X to the construction of an Ulrich sheaf for a finite map of curves, which is in turn equivalent to a higher-rank Brill-Noether problem for any of a certain class of curves on X. Then we show that existence of an Ulrich sheaf for a finite map of curves implies sharp numerical constraints involving the degree of the map and the ramification divisor.Comment: 12 pages, comments welcome; (v3) minor improvement