Augmentations, reduced ideal point gluings and compact type degenerations of curves
Valery Alexeev, Alexander Kuznetsov
TL;DR
This work analyzes two geometric gluings of enhanced triangulated categories—augmentations of curves and ideal-point gluings of pairs of curves—and uncovers exotic exceptional objects arising in these contexts. It develops a detailed invariant and Serre-functor framework for gluing, showing how Hochschild invariants, K-theory, and Jacobians behave under gluing, and demonstrates that BN-exceptional objects yield meaningful BN-modifications of curve-derived categories. A key contribution is linking these gluings to compact-type degenerations of curves by constructing a smooth, proper family of categories whose general fiber is an augmentation and whose central fiber is the reduced ideal-point gluing, using categorical absorption of singularities KS24. This provides a categorical perspective on degenerations and offers a toolkit for studying noncommutative/derived-geometric aspects of curves and their moduli, including connections to Clifford-algebra-based descriptions and square-root stack realizations.
Abstract
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.