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arXiv · 2106.04855

Determinantal Singularities

Anne Frühbis-Krüger·Matthias Zach
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Abstract

We survey determinantal singularities, their deformations, and their topology. This class of singularities generalizes the well studied case of complete intersections in several different aspects, but exhibits a plethora of new phenomena such as for instance non-isolated singularities which are finitely determined, or smoothings with low connectivity; already the union of the coordinate axes in $(\mathbb{C}^3,0)$ is determinantal, but not a complete intersection. We start with the algebraic background and then continue by discussing the subtle interplay of unfoldings and deformations in this setting, including a survey of the case of determinantal hypersurfaces, Cohen-Macaulay codimension $2$ and Gorenstein codimension $3$ singularities, and determinantal rational surface singularities. We conclude with a discussion of essential smoothings and provide an appendix listing known classifications of simple determinantal singularities.

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Anne Frühbis-Krüger, Matthias Zach. 2021-06-09. Determinantal Singularities. https://arxiv.org/abs/2106.04855

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