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2. Exceptional collections

    1. HMS for ATVs (e.g. $x^2+y^2+z^2=0/\mathbb{R}$)


      Add remark

      Problem 2.1.

      Does the derived category of an arithmetic toric variety (i.e., one defined over a non-algebraically closed field) admits an exceptional collection? (e.g., $\{x^2+y^2+z^2=0\}$ in $\mathbb{P}^2_{\mathbb{R}}$)
      • $\bullet$ Over $\mathbb{C}$ is just $\mathbb{P}^1$ but has no $\mathbb{R}$-points
      • $\bullet$ has exceptional collection $<\mathcal{O},\mathcal{E}>$ with $End(\mathcal{E})=\mathbb{H}$.
      • $\bullet$ has an action of $S^1$ which is the restriction of the $S^1\subset \mathbb{C}^*$ action on $\mathbb{C}-points)
      • $\bullet$ toric data here is roughly data of a fan + data of a $Gal(\overline{k}/k)$ action on the fan + ...
      • $\bullet$ One source: An exceptional collection where fan automorphism permutes objects; e.g. powers of toric Frobenius applied to $\mathcal{O}_X$ (gives generating objects acted on by $Gal$, sometimes exceptional collection).


        • Add remark

          Problem 2.2.

          Does every smooth toric variety admits an exceptional collection of sheaves
          • $\bullet$ if replace sheaves (good objects of $D^b(X)$, answer is yes)
          • $\bullet$ First open case is the case of toric 3-folds
          • $\bullet$ if replace sheaves with line bundles, answer is no [Etimov ’14] counterexample smooth Fano toric with $dim\geq 0$ and $Pic=3$.
          • $\bullet$ For surfaces can do with line bundles, read off explicitly from fan (exposition of this in [Hacking-Keating])
          • $\bullet$ More specifically, is it possible to find an exceptional collection from (push-forwards of) line bundles on toric strata? (This case can be analyzed, perhaps).

              Cite this as: AimPL: Syzygies and mirror symmetry, available at http://aimpl.org/syzygyms.