Louis Esser (UCLA)

Tuesday, March 1, 2022 - 2:00pm

Zoom

**Preseminar 2-2:30**

**Title**: The geography of general type varieties

**Abstract**: The classification of algebraic varieties (e.g., compact complex manifolds) is one of the main goals of algebraic geometry. This talk will describe the starting point of this classification, with a focus on the most common class of varieties: general type. Using geometric invariants, we can produce a "map" of all possible such varieties. We'll consider how to chart the edges of this map and see which regions are really "populated" by genuine examples

**Seminar 2:30-3:30**

**Title**: Varieties of general type with large canonical dimension

**Abstract**: By a theorem of Hacon–McKernan, Takayama, and Tsuji, for every n there is a constant r_n for which every smooth variety X of dimension n of general type has birational pluricanonical maps |mK_X| for m > r_n.

The optimal values of r_n remain unknown in dimensions n ≥ 3.

In this work, we calculate exactly the analogous optimal bounds r_n,n and r_n,n-1 for varieties of general type of any dimension n and canonical dimension n, and n-1, respectively. We also determine the optimal lower bound on volume for each of these two classes of varieties.

This talk is based on joint work with Meng Chen and Chengxi Wang, arXiv link: https://arxiv.org/abs/

__2201.08966__

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