Iskovskih seminar in Steklov Institute

November 2019, Moscow

Title: The variety of semistable del Pezzo surfaces

Abstract:
Kollar introduced a generalization of GIT stability for hypersurfaces: stability over rings. This notion is useful for finding good reductions to finite characterstic and for finding good (semistable) birational models of fibrations. I will talk about extending this notion to del Pezzo surfaces of degree 1 and 2. These surfaces are hypersurfaces in weighted projective spaces, where GIT techniques do not work. On the other hand embeddings into the projective spaces are not complete intersections and are difficult to work with. I will talk about generalizations of stability in both of these settings. In particular, I will talk about parameter spaces of del Pezzo surfaces of degrees 1 and 2

Workshop on Algebraic Geometry in Gunsan

October 2018, Gunsan

Title: Stability of del Pezzo surfaces over rings

Abstract:
Kollar introduced a generalization of GIT stability for hypersurfaces: stability over rings. This notion is useful for finding good reductions to finite characterstic and for finding good (semistable) birational models of fibrations. I will talk about extending this notion to del Pezzo surfaces of degree 1 and 2. These surfaces are hypersurfaces in weighted projective spaces, where GIT techniques do not work. On the other hand embeddings into the projective spaces are not complete intersections and are difficult to work with. I will talk about generalizations of stability in both of these settings. In particular, I will talk about parameter spaces of del Pezzo surfaces of degrees 1 and 2.

Seminar in KIAS

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.