Stability conditions on stacks, III

Mini-Corsi di Dottorato, Lezione III Many geometric structures on algebraic varieties (such as vector bundles or coherent sheaves) are best studied by considering the collection of all such structures, modulo some natural equivalence, and giving it a geometric structure itself. Depending on the moduli problem considered, this may lead to a moduli scheme, a moduli space, or a moduli stack. Focusing on examples related to vector bundles on smooth curves, we will discuss the geometry of the corresponding moduli spaces and stacks, explaining how the notion of stability throws a bridge from stacks to spaces. This will be preceded by some relevant background (though very informal and example-driven) on stacks and on geometric invariant theory.