Abstract: Ulrich bundles are certain arithmetically Cohen Macaulay bundles on projective varieties enjoying special cohomological properties. Their study originates in a paper by B. Ulrich in 1984 in the framework of commutative algebra and the attention of algebraic geometers was drawn by a landmark paper by Eisenbud-Schreyer-Weyman, where, among other things, the Chow form of a projective variety X is computed using Ulrich bundles on X, if they exist. It is furthermore conjectured that any projective variety should carry an Ulrich bundle. This conjecture remains widely open in general, as well as related questions regarding possible ranks of Ulrich bundles, their stability and moduli, although a lot is known for specific classes of varieties (e.g., curves, Segre and Grassmann varieties, rational scrolls, complete intersections, some classes of surfaces and threefolds, etc.)
In the talk I will present recent results concerning existence, ranks, stability and moduli of Ulrich bundles on Fano threefolds, obtained in collaboration with C. Ciliberto and F. Flamini (arXiv:2206.09986 and arXiv:2205.13193), extending and unifying results by (many) other authors.
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