Abstract
Determinantal singularities are an important class of singularities, generalizing complete intersections, which recently have seen a large amount of interest. They are defined as preimage of \(M^{ t }_{m,n}\) the sets of matrices of rank less than t. The rank stratification of \(M^{ t }_{m,n}\) gives rise to some interesting structures on determinantal singularities. In this article we will focus on one of these, namely the Tjurina transform. We will show some properties of it, and discuss how it can or cannot be used to find resolutions of determinantal singularities.
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Notes
- 1.
It is of course also possible that p −1(0) is a irreducible component of \(X\times _F\operatorname {Tjur}(M^{ t }_{m,n})\) even without the condition on the dimensions, we will discuss this later in Proposition 4.3.
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Acknowledgements
I wish to thank Maria Ruas for introducing me to the subject of determinantal singularities and for many fruitful conversations during the preparation of this article, and to thank Bárbara Karolline de Lima Pereira who found many mistakes in the earlier version of the article while writing her Master thesis.
The author was supported by FAPESP grant 2015/08026-4.
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Pedersen, H.M. (2021). On Tjurina Transform and Resolution of Determinantal Singularities. In: Fernández de Bobadilla, J., László, T., Stipsicz, A. (eds) Singularities and Their Interaction with Geometry and Low Dimensional Topology . Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-61958-9_13
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