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 c
onstruct an exceptional collection on a nonnormal semistable degeneration of a 1-parameter family of Burniat surfaces with $K^2=6$.
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 $K_0(mathcal A)=bZ_2^6$.
It was conjectured by McKernan and Shokurov that for all Mori contractions from X to Y of given dimensions, for any positive epsilon there is a positive delta, such that if X is epsilon-log terminal, then Y is delta-log terminal. We prove this conjec
ture in the toric case and discuss the dependence of delta on epsilon, which seems mysterious.
It was conjectured in cite{Namikawa_ExtendedTorelli} that the Torelli map $M_gto A_g$ associating to a curve its jacobian extends to a regular map from the Deligne-Mumford moduli space of stable curves $bar{M}_g$ to the (normalization of the) Igusa b
lowup $bar{A}_g^{rm cent}$. A counterexample in genus $g=9$ was found in cite{AlexeevBrunyate}. Here, we prove that the extended map is regular for all $gle8$, thus completely solving the problem in every genus.
