broadcast in Room 515, Cosmology Bldg., NTU, Online seminar
(線上演講 於台大次震宇宙館515教室直播/收播)
On Pluricanonical Maps of Varieties of General Type
Justin Lacini (Kansas University)
Abstract:
Hacon and McKernan have proved that there exist integers
![](https://chart.googleapis.com/chart?cht=tx&chl=%24r_%7Bn%7D%24&chf=bg,s,333333&chco=ffffff)
such that if
![](https://chart.googleapis.com/chart?cht=tx&chl=%24X%24&chf=bg,s,333333&chco=ffffff)
is a smooth variety of general type and dimension
![](https://chart.googleapis.com/chart?cht=tx&chl=%24n%24&chf=bg,s,333333&chco=ffffff)
, then the pluricanonical maps
![](https://chart.googleapis.com/chart?cht=tx&chl=%24%7CrKX%7C%24&chf=bg,s,333333&chco=ffffff)
are birational for
![](https://chart.googleapis.com/chart?cht=tx&chl=%24r%20%5Cgeq%20r_%7Bn%7D%24&chf=bg,s,333333&chco=ffffff)
. These values are typically very large: for example
![](https://chart.googleapis.com/chart?cht=tx&chl=%24r_%7B3%7D%20%5Cgeq%2027%24&chf=bg,s,333333&chco=ffffff)
and
![](https://chart.googleapis.com/chart?cht=tx&chl=%24r_%7B4%7D%20%5Cgeq%2094%24&chf=bg,s,333333&chco=ffffff)
. In this talk we will show that the
![](https://chart.googleapis.com/chart?cht=tx&chl=%24r%24&chf=bg,s,333333&chco=ffffff)
-th canonical maps of smooth threefolds and fourfolds of general type have birationally bounded fibers for
![](https://chart.googleapis.com/chart?cht=tx&chl=%24r%20%5Cgeq%202%24&chf=bg,s,333333&chco=ffffff)
and
![](https://chart.googleapis.com/chart?cht=tx&chl=%24r%20%5Cgeq%204%24&chf=bg,s,333333&chco=ffffff)
respectively. Furthermore, we will generalize these results to higher dimensions in terms of the constants
![](https://chart.googleapis.com/chart?cht=tx&chl=%24r_%7Bn%7D%24&chf=bg,s,333333&chco=ffffff)
and we will discuss recent progress on a conjecture of Chen and Jiang.
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