##### Department of Mathematics,

University of California San Diego

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### Algebraic Geometry Seminar

## Roberto Svaldi

#### MIT

## On Fano varieties appearing as fibers of a Mori fiber space.

##### Abstract:

Mori fiber spaces (MFS) are one of the building blocks in the Minimal Model Program. These are maps $X \to Y$ between normal varieties with nice singularities, such that $\dim Y <\dim X$, $\rho(X/Y)=1$ and $-K_X$ is ample on every fiber. In particular, most fibers will be $Q$-Fano varieties. Starting from classical results on the topology of fibrations, I will try to explain how the above conditions place strong restrictions on what varieties can appear as fibers of MFS. I will give characterizations for low-dimensional varities and explain what happens in the toric category. Moreover, we will show that this question can be connected to the question of existence of Kaehler-Einstein metrics. Joint work with G. Codogni, A. Fanelli, L. Tasin.

Host: James McKernan

### May 23, 2014

### 2:30 PM

### AP&M 7218

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