An exceptional object for the irregular Waldron--Witaszek Fano fourfold

Today's arXiv update includes Joe Waldron and Jakub Witaszek's paper, “An irregular smooth Fano fourfold in positive characteristic”. Here, irregular means that $\mathrm{H}^1(X,\mathcal{O}_X)\neq 0$. This is not possible for Fano varieties in characteristic zero, by Kodaira vanishing. But the authors construct a Fano fourfold over the algebraic closure of $\mathbb{F}_2$ with $\mathrm{H}^1(X,\mathcal{O}_X)\cong\overline{\mathbb{F}}_2$. Whilst this is very interesting for questions about rationality, I am always thinking about derived categories. And, unlike for Fano varieties in characteristic zero, the isomorphism $\mathrm{H}^1(X,\mathcal{O}_X)\cong\overline{\mathbb{F}}_2$ implies that $\mathcal{O}_X$ (or for that matter, any line bundle!) is not an exceptional object. This is unfortunate because we like to start decomposing derived categories of Fano varieties by considering an exceptional sequence of line bundles and then continuing from there. So what's up with the derived category…

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