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Augmentations, reduced ideal point gluings and compact type degenerations of curves

In this note we demonstrate some unexpected properties that simple gluings of the simplest derived categories may have. We consider two special cases: the first is an augmented curve, i.e., the gluing of the derived categories of a point and a curve with the gluing bimodule given by the structure sheaf of the curve; the second is an ideal point gluing of curves, i.e., the gluing of the derived categories of two curves with the gluing bimodule given by the ideal sheaf of a point in the product of the curves. We construct unexpected exceptional objects contained in these categories and discuss their orthogonal complements. We also show that the simplest example of compact type degeneration of curves, a flat family of curves with a smooth general fiber and a 1-nodal reducible central fiber, gives rise to a smooth and proper family of triangulated categories with the general fiber an augmented curve and the central fiber the orthogonal complement of the exotic exceptional object in the ideal point gluing of curves, called the reduced ideal point gluing of curves.