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We show how some of the refined tropical counts of Block and Göttsche emerge from the wall-crossing formalism. This leads naturally to a definition of a class of putative $q$-deformed Gromov–Witten invariants. We prove that this coincides with another natural $q$-deformation, provided by a result of Reineke and Weist in the context of quiver representations, when the latter is well defined.
Let $N$ be a point of an orbit closure $\overline{{\mathcal O}}_M$ in a module variety such that its orbit ${\mathcal O}_N$ has codimension 2 in $\overline{{\mathcal O}}_M$. We show that under some additional conditions the pointed variety $(\overline{{\mathcal O}}_M,N)$ is smoothly equivalent to a cone over a rational normal curve.
This paper investigates how to relate the syzygy periodicity of a self-injective algebra $A$ to its Auslander–Reiten periodicity. Moreover, a characterization is provided of the Auslander–Reiten bounded $A$–$A$-bimodules that are periodic.
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