## Workshop in algebraic geometry

**Speakers:***Maksym Fedorchuk (Boston College) *Antonella Grassi (University of Pennsylvania) *Yuchen Liu (Yale University) *Xiaowei Wang (Rutgers University -- Newark)

**Schedule:**

9:00 - 9:30 | Coffee | Room: Joseph Henry Room (Jadwin) |

9:30 - 10:30 | Yuchen Liu
Local volume of klt singularities and application to moduli problems |
Room: Joseph Henry Room (Jadwin) |

10:45 - 11:45 | Xiaowei Wang
On the proper moduli space of smoothable K-semistable Fano varieties |
Room: Joseph Henry Room (Jadwin) |

12:00 - 2:00 | Lunch | Room: Professor's Lounge (PL) - Fine Hall |

2:15 - 3:15 | Maksym Fedorchuk
Standard models of low degree del Pezzo fibrations via GIT for Hilbert points |
Room: Joseph Henry Room (Jadwin) |

3:30 - 4:15 | Tea | Math Department Common Room (3rd floor) |

4:15 - 5:15 | Antonella Grassi
Some questions on Local, Global and Local to Global properties, in algebraic geometry, topology and strings |
Room: Joseph Henry Room (Jadwin) |

## Abstracts:

**Yuchen Liu**: Motivated from work in differential geometry, Chi Li introduced the local volume of a klt singularity as the minimum normalized volume of all valuations centered at this singularity. The local volume turns out to play an important role in the study of explicit proper moduli space of K-polystable Fano varities. If the global volume is not so small, then very often the local volume gives a strong control on the singularities appearing at boundary objects of the moduli space. In this talk we will discuss properties of the local volume and illustrate how it can be used to study explicit moduli spaces, such as cubic surfaces and cubic threefolds. Partly based on joint work with Chenyang Xu.

**Xiaowei Wang**: In this talk I will explain our construction (joint with Chi Li and Chenyang Xu) of the proper moduli space of Q-Gorenstein smoothable K-semistable Fano varieties.

**Maksym Fedorchuk**: A del Pezzo fibration is one of the natural outputs of the Minimal Model Program for threefolds. At the same time, geometry of an arbitrary del Pezzo fibration can be unsatisfying due to the presence of non-integral fibers and terminal singularities of an arbitrarily large index. In 1996, Corti developed a program of constructing `standard models' of del Pezzo fibrations within a fixed birational equivalence class. Standard models enjoy a variety of desired properties, one of which is that all of their fibers are Q-Gorenstein integral del Pezzo surfaces. Corti proved the existence of standard models for del Pezzo fibrations of degree d\geq 2, with the case of d=2 being the most difficult. The case of d=1 remained a conjecture. In 1997, Kollár recast and improved the Corti's result in degree d=3 using ideas from Geometric Invariant Theory for cubic surfaces. I will present a generalization of Kollár's approach in which we develop notions of stability for families of low degree (d\leq 2) del Pezzo fibrations in terms of their Hilbert points (i.e., low degree equations cutting out del Pezzos). A correct choice of stability and a bit of enumerative geometry then leads to (very good) standard models in the sense of Corti. This is joint work in progress with Hamid Ahmadinezhad and Igor Krylov.

**Antonella Grassi**: TBA

We are thankful for support from the Princeton mathematics department.

**Contact**: kascher@princeton.edu