##### Department of Mathematics,

University of California San Diego

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### Math 208 - Algebraic Geometry Seminar

## Justin Lacini

#### UCSD

## On pluricanonical maps of varieties of general type

##### Abstract:

Hacon and McKernan have proved that there exist integers $r_n$ such that if $X$ is a smooth variety of general type and dimension $n$, then the pluricanonical maps $|rK_X|$ are birational for all $r\geq r_n$. These values are typically very large: for example $r_3\geq 27$ and $r_4\geq 94$. In this talk we will show that the $r^{\textup{th}}$ canonical maps of smooth threefolds and fourfolds of general type have birationally bounded fibers for $r\geq 2$ and $r\geq 4$ respectively. Furthermore, we will generalize these results to higher dimensions in terms of the constants $r_n$ and we will discuss recent progress on a conjecture of Chen and Jiang.

Host: James McKernan

### October 25, 2019

### 1:45 PM

### AP&M 7321

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