The subregular variety to the variety of special lattices
Let \(k\) be the algebraic closure of a finite field of characteristic \(p\) and let \(\mathcal{O}\) be the \(k\)-points of the Witt vectors and \(K\) the fraction field of \(\mathcal{O}\). Let \(F=\mathcal{O}^n\subset K^n\) be the standard inclusion. A lattice \(L\) as usual will be a free \(\mathcal{O}\)-submodule of rank \(n\) of \(K^n\). It is called special if \(\bigwedge^n L=\bigwedge^n F\) and it is of height at most \(r>0\) if \(L\subset p^{-r}F\). \textit{W. J. Haboush} proved that the set of all special lattices (fixing \(n,r\)) form a projective variety [Tohoku Math. J. (2) 57, No. 1, 65--117 (2005; Zbl 1119.14004)]. In the author's thesis, he proved that these are normal and the author and Haboush proved that these are local complete intersections. In the present article, the author shows that the singular locus of the lattice variety is also normal and local complete intersection.
scientific article; zbMATH DE number 5833923