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Divisors on Burniat surfaces

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 Added by Valery Alexeev
 Publication date 2013
  fields
and research's language is English




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In this short note, we extend the results of [Alexeev-Orlov, 2012] about Picard groups of Burniat surfaces with $K^2=6$ to the cases of $2le K^2le 5$. We also compute the semigroup of effective divisors on Burniat surfaces with $K^2=6$. Finally, we construct an exceptional collection on a nonnormal semistable degeneration of a 1-parameter family of Burniat surfaces with $K^2=6$.



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We construct an exceptional collection $Upsilon$ of maximal possible length 6 on any of the Burniat surfaces with $K_X^2=6$, a 4-dimensional family of surfaces of general type with $p_g=q=0$. We also calculate the DG algebra of endomorphisms of this collection and show that the subcategory generated by this collection is the same for all Burniat surfaces. The semiorthogonal complement $mathcal A$ of $Upsilon$ is an almost phantom category: it has trivial Hochschild homology, and $K_0(mathcal A)=bZ_2^6$.
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