We discuss the minimal model program for b-log varieties, which is a pair of a variety and a b-divisor, as a natural generalization of the minimal model program for ordinary log varieties. We show that the main theorems of the log MMP work in the setting of the b-log MMP. If we assume that the log MMP terminates, then so does the b-log MMP. Furthermore, the b-log MMP includes both the log MMP and the equivariant MMP as special cases. There are various interesting b-log varieties arising from different objects, including the Brauer pairs, or “non-commutative algebraic varieties which are finite over their centres”. The case of toric Brauer pairs is discussed in further detail.

We classify the solutions to a system of equations, introduced by Bondal, that encode numerical constraints on full exceptional collections of length 4 on surfaces. The corresponding result for length 3 is well-known and states that there is essentially one solution, namely the one corresponding to the standard exceptional collection on the surface \(\mathbb{P}^2\). This was essentially proven by Markov in 1879. It turns out that in the length 4 case, there is one special solution which corresponds to \(\mathbb{P}^1 \times \mathbb{P}^1\) whereas the other solutions are obtained from \(\mathbb{P}^2\) by a procedure we call numerical blowup. Among these solutions, three are of geometric origin: \(( \mathbb{P}^2 \cup \{\cdot \})\), \(\mathbb{P}^1 \times \mathbb{P}^1\) and the ordinary blowup of \(\mathbb{P}^2\) at a point. The other solutions are parametrized by N and very likely do not correspond to commutative surfaces. However they can be realized as noncommutative surfaces, as was recently shown by Dennis Presotto and the first author

In this note we consider a notion of relative Frobenius pairs of commutative rings \(S/R\). To such a pair, we associate an \(\mathbb{N} \)-graded \(R\)-algebra \(\prod_R(S) \) which has a simple description and coincides with the preprojective algebra of a quiver with a single central node and several outgoing edges in the split case. If the rank of \(S\) over \(R\) is 4 and \(R\) is noetherian, we prove that \(\prod_R(S)\) is itself noetherian and finite over its center and that each \( \prod_R(S)_d \) is finitely generated projective. We also prove that \(\prod_R(S)\) is of finite global dimension if \(R\) and \(S\) are regular.

We relate the deformation theory of Ginzburg Calabi-Yau algebras to negative cyclic homology. We do this by exhibiting a DG-Lie algebra that controls this deformation theory and whose homology is negative cyclic homology. We show that the bracket induced on negative cyclic homology coincides with Menichi’s string topology bracket. We show in addition that the obstructions against deforming Calabi-Yau algebras are annihilated by the map to periodic cyclic homology. In the commutative case we show that our DG-Lie algebra is homotopy equivalent to \((T_{\textrm{poly}}[[u]],−u\textrm{div})\).

We give concrete DG-descriptions of certain stable categories of maximal Cohen-Macaulay modules. This makes it possible to describe the latter as generalized cluster categories in some cases.

We prove a technical result which allows us to establish the non- degeneracy of potentials on quivers in some previously unknown or non-obvious cases. Our result applies to certain McKay quivers and also to potentials derived from geometric helices on Del Pezzo surfaces.