A note on nodal determinantal hypersurfaces (Q2196644)

It is important in modern algebraic geometry to have tools to be able to construct examples of varieties with mild singularities. In this paper, the author presents a result that guarantees that some determinantal varieties have good singularities. More precisely, given a morphism of vector bundles \(\sigma: \mathcal{E}\rightarrow \mathcal{F}\) of rank \(n\) on a smooth variety, we define the degeneracy locus \(D_r(\sigma)\) as the locus where the rank of sigma is less or equal than \(r\). Assuming that the degeneracy places form a stratification in smooth varieties of expected dimension for \(r=0,\dots, n\), and the dimension of the variety is \(4\), the author shows that the determinantal variety is nodal provided that it is irreducible. This allows the author to construct examples of 3-dimensional Calabi-Yau and Fano varieties with only ODP singularities. To obtain this result, the author studies what happens to the Chern classes of determinantal varieties under the \(n\)-generality hypothesis of \(\sigma\). He also proves a very explicit formula for intersection homology Euler characteristic for these determinantal varieties. There are many explicit examples in the paper and the prerequisites are clearly explained.