Abstract. Let G be a polyhedral group G ⊂ SO(3) of types Z/nZ, D2n and T. We prove that there exists a one-to-one correspondence between flops of G-Hilb(C 3 ) and mutations of the McKay quiver with potential which do not mutate the trivial vertex. This correspondence provides two equivalent methods to construct every projective crepant resolution for the singularities C 3 /G, which are constructed as moduli spaces MC of quivers with potential for some chamber C in the space Θ of stability conditions. In addition, we study the relation between the exceptional locus in MC with the corresponding quiver QC, and we describe explicitly the part of the chamber structure in Θ where every such resolution can be found.
Abstract. In this paper we consider the iterated G-equivariant Hilbert scheme G/N -Hilb(NHilb) and prove that G/N -Hilb(N -Hilb(C 3 )) is a crepant resolution of C 3 /G isomorphic to the moduli space M θ (Q) of θ-stable representations of the McKay quiver Q for certain stability condition θ. We provide several explicit examples to illustrate this construction. We also consider the problem of when G/N-Hilb(N-Hilb) is isomorphic to G-Hilb showing the fact that these spaces are most of the times different.
This study presents an experience that combines problem-posing and Math Trails in the context of future teachers’ instruction. Pre-service teachers in the third year of their studies were faced with the design of tasks to be included in Math Trails for primary school students. The study analyzes, from a quantitative approach, 117 tasks contained in 11 Math Trails. The analysis was performed on the basis of classification variables (grade, mathematical content and object or real element involved in every task) and research variables which provide information about the nature of the tasks (procedural vs. problem-solving, level of cognitive demand, degree of contextualization, openness and creativity). Additionally, relationships between the different categories of analysis have been studied. The results reveal certain biases in the tasks in relation to the contents addressed (an abundance of tasks with a geometric component and a scarcity of tasks involving algebra or probability concepts). Most of the tasks are presented in a real context. However, a higher percentage of procedural tasks than problem-solving tasks is observed, with a predominance of low openness, creativity and cognitive demand. These results provide useful lines of work to address difficulties faced by future teachers in the STEAM field.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.