##### Department of Mathematics,

University of California San Diego

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### Algebraic Geometry

## Omprokash Das

#### UCLA

## Birational geometry of Surfaces and 3-folds over Imperfect Fields

##### Abstract:

Lots of progress have been made in the recent years on the birational geometry of surfaces and 3-folds in positive characteristic over algebraically closed field. The same can not be said about the varieties over imperfect fields. These varieties appear naturally in positive characteristic while studying fibrations (as a generic fiber). Recently the minimal model program (MMP) for surfaces over excellent base scheme was successfully carried out by Tanaka. He also showed that the abundance conjecture holds for surfaces over imperfect fields. His results have become one of main tools for studying fibrations in positive characteristic. One of the things that is not covered in Tanaka's papers is the del Pezzo surfaces (a regular surface with -$K_X$ ample) over imperfect fields. One interesting feature of del Pezzo surfaces is that over an algebraically closed field they satisfy the Kodaira vanishing theorem. This makes the theory of del Pezzo surfaces quite interesting. However, over imperfect fields it was known for a while that in char 2, Kodaira vanishing fails for del Pezzo surfaces, due to (Schroer and Maddock). It is only very recently that some positive results started to show up. In a recent paper by Patakfalvi and Waldron it was shown that the Kodaira vanishing theorem holds for del Pezzo surfaces over imperfect fields in char $p>3$. In this talk I will show that in fact the Kawamata-Viehweg vanishing theorem holds for del Pezzo surfaces over imperfect fields in char $p>3$. I will also report on a project which is a work in progress (joint with Joe Waldron) on the minimal model program for 3-folds over imperfect fields and the BAB conjecture for del Pezzo surfaces over imperfect fields.

James KcKernan

### October 6, 2017

### 2:30 PM

### AP&M B412

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