This paper studies Batyrev's notion of primitive collection. We use primitive collections to characterize the nef cone of a quasi-projective toric variety whose fan has convex support, a result stated without proof by Batyrev in the smooth projective case. When the fan is non-simplicial, we modify the definition of primitive collection and explain how our definition relates to primitive collections of simplicial subdivisons. The paper ends with an open problem.2000 Mathematics Subject Classification. Primary 14M25

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Improving college STEM (science, technology, engineering, mathematics) student learning outcomes is an ongoing area of focus in Institutions of Higher Education (IHE). This reform includes challenging, changing, and adapting both teaching practices and the learning environment. Communities of practice (CoPs) can support faculty in making these shifts; however, creating large-scale instructional changes in STEM education requires a more careful look at the existing systems and structures in place. In this paper, we investigate a network of regional CoPs composed mainly of mathematics faculty from IHE focused on teaching with inquiry methods. Understanding what faculty need and value to support their instructional changes is important as CoPs and other mechanisms are put in place to increase student success. In this qualitative study, we use the value framework developed by Wenger et al. ( 2011 ) to dissect the variety of ways faculty engage and find value in their CoP participation. Faculty participants expressed that CoP participation created unique layers of value in helping them to identify resources to support teaching with inquiry especially during a pandemic, shift their beliefs about teaching, and engage with a network of peers about mathematics and teaching. Findings from this study, conducted during the COVID-19 global pandemic, provide preliminary insights for STEM stakeholders interested in large-scale, ongoing instructional reform to improve student learning outcomes and for networks interested in collectively supporting CoPs with ongoing rather than finite goals. Supplementary Information The online version contains supplementary material available at 10.1007/s41979-021-00061-3.

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