Abstract: Given a smooth homogeneous polynomial, how to determine if it is the determinant of a matrix of linear forms? The root of this problem dates back to the nineteenth century, and to our best knowledge, it was first stated clearly by Hesse in 1855. As it turns out, the problem is equivalent to the existence of certain objects, known as Ulrich sheaves of rank one on the hypersurface determined by the homogeneous polynomial. Ulrich bundles first appeared in the realm of Commutative Algebra through the work of B. Ulrich in the 1980s, and paved its way into the realm of Algebraic Geometry through the work of D. Eisenbud and F.O. Schreyer in the early 2000s. In this talk, we will go through a brief historical overview of Ulrich bundles followed by some of my recent results on the classification of varieties with naturally associated Ulrich bundles, obtained in collaboration with A.F. Lopez, V. Antonelli and G. Casnati.
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