|Title: Derived Categories, Arithmetic, and Rationality Questions|
|Speaker: Alicia Lamarche of University of South Carolina|
|Contact: David Zureick-Brown, email@example.com|
|Date: 2019-10-08 at 4:00PM|
|Venue: MSC W303|
When trying to apply the machinery of derived categories in an arithmetic setting, a natural question is the following: for a smooth projective variety $X$, to what extent can $D^b(X)$ be used as an invariant to answer rationality questions? In particular, what properties of $D^b(X)$ are implied by $X$ being rational, stably rational, or having a rational point? On the other hand, is there a property of $D^b(X)$ that that implies that $X$ is rational, stably rational, or has a rational point? \\ In this talk, we will examine a family of arithmetic toric varieties for which a member is rational if and only if its bounded derived category of coherent sheaves admits a full \'etale exceptional collection. Additionally, we will discuss the behavior of the derived category under twisting by a torsor, which is joint work with Matthew Ballard, Alexander Duncan, and Patrick McFaddin.
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