Modularity of the Minimal Model Programme

A run of the Minimal Model Programme for a projective variety consists of a series of birational modifications which simplifies the variety. One of the most important results in algebraic geometry is that, under suitable conditions, it is possible to make such a run. When the variety under investigation is a moduli space, one can obtain birational modifications also from modifying the class of objects parametrized by the moduli space. For instance, a birational modification of the moduli space of nodal curves can be obtained by relaxing the condition on the singularities, and let the moduli space parametrize also curves with cusps. We call this sort of modifications “modular modifications".
In this talk, I will discuss cases where a run of the Minimal Model Programme can be described in terms of modular modifications. More specifically, I will discuss the moduli space of curves (joint work with L. Tasin and F. Viviani) and the moduli space of rank two vector bundles over a del Pezzo surface of degree one (joint work with C. Casagrande and A. Fanelli).