August 2018, Seoul

**Title:** Semistability of del Pezzo surfaces, good models and reductions

**Abstract:**

Kollar has introduces a notion of stability of a hypersurface in projective space over a ring. This notion can be used to find good reductions of hypersurfaces to finite characteristic. It also has application to birational geometry. If a cubic fibration over an affine curve is a semistable cubic surface over the coordinate ring of the base, then the fibration is a Mori fiber space and the total space has only Gorenstein singularities. This is an improvment over an existing result of Corti, whose version gave models with also Gorenstein but worse singularities. On the other hand, Corti’s method allowed to prove similar result for del Pezzo fibrations of degree 2. I will talk about extending the results of Kollar to del Pezzo surfaces of degree 2 and 1. As a corollary we improve Corti’s result and prove an analogue in degree 1.