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Determinantal Singularities

In: Handbook of Geometry and Topology of Singularities IV

Author

Listed:
  • Anne Frühbis-Krüger

    (Carl-von-Ossietzky Universität Oldenburg)

  • Matthias Zach

    (Leibniz Universität Hannover)

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 ( ℂ 3 , 0 ) $$({\mathbb C}^3,0)$$ 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.

Suggested Citation

  • Anne Frühbis-Krüger & Matthias Zach, 2023. "Determinantal Singularities," Springer Books, in: José Luis Cisneros-Molina & Lê Dũng Tráng & José Seade (ed.), Handbook of Geometry and Topology of Singularities IV, chapter 0, pages 45-159, Springer.
    • Handle: RePEc:spr:sprchp:978-3-031-31925-9_2
    DOI: 10.1007/978-3-031-31925-9_2
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