The study of vector bundles on algebraic varieties is a classical tópico of researchwithin algebraic geometry. The present proposal focuses principally on two classes of bundles: Steiner bundles and instanton bundles.For the first class of bundles, we consider the classification of Steiner bundles on projective varieties. In order to do so, we will try to generalize the definition of Schwarzenberger bundle for any rank on a generic projective variety and will also try to generalize the concept of jumping pair for a Steiner bundle. In the cases which have been already studied(projective spaces and Grassmannians) the locus of the jumping pairs has been the key that allowed to classify Steiner bundles, with locus of maximal dimension, and classify them as Schwarzenberger. Our belief is that it will be sufficient to give all definitions for a specific variety and that the general case will be recovered from this one.Regarding instanton bundles, the problem of the defining them for varieties other thanprojective spaces has been of great concern recently. We will consider the particular case ofGrassmannians, setting the definitions and studying the first properties, that willinclude their stability and the description of their moduli spaces.
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