김선정 (경상대학교)

김윤환 (서울대학교)

김신영 (IBS)

- Title: On the deformation rigidity of smooth projective symmetric varieities with Picard number one

- Abstract: Symmetric varieties are normal equivariant open embeddings of symmetric homogeneous spaces and they are interesting examples of spherical varieties. The principal goal of this article is to study the rigidity under Kahler deformations of smooth projective symmetric varieties with Picard number one. This is a joint work with K.-D. Park.

양윤정 (충남대학교)

조용화 (KIAS)

- Title: Degeneration of del Pezzo surfaces: a categorical viewpoint

- Abstract: For a full exceptional collection in a del Pezzo surface, one may associate a toric surface which is a Q-Gorenstein degeneration of the given del Pezzo surface. Conversely, if we are given a Q-Gorenstein smoothing of a toric surface to a del Pezzo surface, one may construct a full exceptional collection on the general fiber. However, a del Pezzo surface may have several exceptional collections which are "numerically equivalent", and accordingly, admit the isomorphic toric surfaces. Moreover, it can be shown that some of these collections cannot be constructed from a single Q-Gorenstein degeneration. One of the natural questions is the following: if we are given two "numerically equivalent" exceptional collections, together with the respective Q-Gorenstein degenerations to the same toric surface, how can we distinguish these two different degenerations? In this talk I will explain my ongoing attempt to answer this question using relative minimal model program.

이용남 (KAIST)

- Title: Smooth specializations and function fields of hypersurfaces in the projective space.

- Abstract: In this presentation, I will briefly talk on smooth specializations of hypersurfaces in a projective smooth manifold, and discuss on dominant rational maps from a very general hypersurface in the projective space.

In the first part of the talk, we give a structure theorem for projective manifolds $W_0$ with the property of admitting a 1-parameter deformation where $W_t$ is a hypersurface in a projective smooth manifold $Z_t$. Their structure is the one of special iterated univariate coverings which we call of normal type. We give an application to the case where $Z_t$ is a projective space, respectively an abelian variety. We also give a characterizaton of smooth ample hypersurfaces in abelian varieties and describe an irreducible connected component of their moduli space. All works in this topic have been carried out by the joint research with Fabrizio Catanese.

In this second part of the talk, we study dominant rational maps from a very general hypersurface $X$ of degree at least $n+3$ in the projective $(n+1)$-space  to smooth projective $n$-folds $Y$ for $nge 3$. It was already proved that $Y$ is a rational surface when $n=2$ by the joint work with Gian Pietro Pirola. Based on Lefschetz theory, Hodge theory, and Cayley-Bacharach property, we prove that there is no dominant rational map from $X$ to $Y$ unless $Y$ is uniruled if the degree of the map is a prime number. Furthermore, we prove that $Y$ is rationally connected when $n=3$ and the degree of the map is a prime number. This work has been carried out by the joint research with De-Qi Zhang.