Seminar series
Date
Tue, 08 Feb 2011
Time
15:45 -
16:45
Location
L3
Speaker
Nicolas Addington
Organisation
Imperial College London
If $X$ is a Fano variety with canonical bundle $O(-k)$, its derived category
has a semi-orthogonal decomposition (I will say what that means)
\[ D(X) = \langle O(-k+1), ..., O(-1), O, A \rangle, \]
where the subcategory $A$ is the "interesting piece" of $D(X)$. In the previous talk we saw that $A$ can have very rich geometry. In this talk we will see a less well-understood example of this: when $X$ is a smooth cubic in $P^5$, $A$ looks like the derived category of a K3 surface. We will discuss Kuznetsov's conjecture that $X$ is rational if and only if $A$ is geometric, relate it to Hassett's earlier work on the Hodge theory of $X$, and mention an autoequivalence of $D(Hilb^2(K3))$ that I came across while studying the problem.