Derived Categories of Fano Threefolds $V_{12}$

In the present paper, we give a description of the derived category of coherent sheaves on a Fano threefold of index 1 and degree 12 (the variety $V_{12}$). It can easily be shown that if $X$ is a $V_{12}$ variety, then its derived category contains an exceptional pair of vector bundles $(\mathscr U,\mathscr O_X)$, where $\mathscr O_X$ is the trivial bundle, and $\mathscr U$ is the Mukai bundle of rank 5 (which induces the embedding $X\to\operatorname{Gr}(5,10)$). The orthogonal subcategory $\mathscr A_X={}^\perp\left<\mathscr U,\mathscr O\right>\subset\mathscr D^b(X)$ can be treated as the nontrivial part of the derived category of $X$. The main result of the present paper is the construction of the category equivalence $\mathscr A_X\cong\mathscr D^b(C^\vee)$, where $C^\vee$ is the curve of genus 7 which can be canonically associated to $X$ according to the results due to Iliev and Markushevich. In the construction of the equivalence, we make use of the geometric results due to Iliev and Markushevich, as well as the Bondal and Orlov results about derived categories. As an application, we prove that the Fano surface of $X$ (which is the surface parametrizing conics on $X$) is isomorphicto $S^2C^\vee$, the symmetric square of the corresponding curve of genus 7.