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Codimension Two Determinantal Varieties with Isolated Singularities

Maria Aparecida Soares Ruas, Miriam Da Silva Pereira

Abstract


We study codimension two determinantal varieties with isolated singularities. These singularities admit a unique smoothing, thus we can define their Milnor number as the middle Betti number of their generic fiber. For surfaces in $\mathsf{C}^4$, we obtain a Lê-Greuel formula expressing the Milnor number of the surface in terms of the second polar multiplicity and the Milnor number of a generic section. We also relate the Milnor number with Ebeling and Gusein-Zade index of the $1$-form given by the differential of a generic linear projection defined on the surface. To illustrate the results, in the last section we compute the Milnor number of some normal forms from Frühbis-Krüger and Neumer [7] list of simple determinantal surface singularities.

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DOI: http://dx.doi.org/10.7146/math.scand.a-19220

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ISSN 0025-5521 (print) ISSN 1903-1807 (online)

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