Nonemptiness of skew-symmetric degeneracy loci (Q819606)

The paper under review deals with nonemptiness results for skew-symmetric degeneracy loci. The main result is the following: Let \(X\) be a rank \(d\)-dimensional complex projective scheme with a rank \(N\) vector bundle \(V\). Suppose that \(V\) is equipped with a skew-symmetric bilinear form with values in a line bundle \(L\) such that the restriction of the form to any fiber has rank at most \(r\), where \(r>0\) is even, and assume that \(\bigwedge^2V^*\otimes L\) is ample. Then, if \(d>2(N-r)\), the locus of points where the rank of the form is at most \(r-2\) is nonempty. The author also gives applications to subschemes of skew-symmetric matrices, and to the stratification of the dual of a Lie algebra by orbit dimension. The method used in this paper are similar to those used in its companion paper [\textit{W. Graham}, Am. J. Math. 127, No. 2, 261--292 (2005; Zbl 1077.14009)].