On the discriminant locus of a rank \(n-1\) vector bundle on \(\mathbb{P}^{n-1}\)
The paper under review is devoted to study geometric properties of the discriminant locus of a vector bundle of rank \(n - 1\) on the projective \((n - 1)\)-space \(\mathbb P^{n-1}\). The discriminant locus \(\Delta(\mathcal E)\) of such a vector bundle \(\mathcal E\) is the subset of \(\mathbb P(H^0(\mathcal E))\) parameterizing the zero schemes of sections of \(\mathcal E\) which are either singular or positive dimensional. In many cases, \(\Delta(\mathcal E)\) is irreducible of codimension one, e.g.\ if \(\mathcal E\) is a direct sum of very ample line bundles and its top Chern class is greater than one. There are, however, indecomposable vector bundles whose discriminant loci are not irreducible or have higher codimensions. For linear systems of divisors, the problem of classifying degenerate discriminant loci was extensively studied in the case of very ample line bundles, while for more general line bundles its study was started by \textit{A. Lanteri} et al. [J. Reine Angew. Math. 477, 199--219, appendix: 216--219 (1996; Zbl 0851.14002)] and continued by \textit{A. Lanteri} and \textit{R. Muñoz} [J. Pure Appl. Algebra 212, No. 4, 808--831 (2008; Zbl 1152.14006); Indiana Univ. Math. J. 58, 2205--2225 (2009; Zbl 1205.14007)]; the classification, however, is still incomplete. As to vector bundles of higher rank, the classification problem has never systematically been explored. The paper under review highlights a large family of vector bundles whose discriminant loci are non-degenerate. More precisely, the main result the author proves, working over an algebraically closed field \(k\) of characteristic zero, is the following. If \(\mathcal E\) is a very ample vector bundle of rank \(n-1\) over \(\mathbb P^{n-1}\) with \(c_{n-1}(\mathcal E) \geq 2\), then \(\Delta(\mathcal E)\) is an irreducible hypersurface whose degree is expressed as an explicit function of invariants of \(\mathcal E\). A remarkable example is offered by Steiner bundles, roughly speaking very ample vector bundles of rank \(n-1\) on \(\mathbb P^{n-1}\) having a minimal free resolution of length one. For each positive twist of such a bundle, the degree of the discriminant hypersurface is reformulated in terms of the graded Betti numbers of \(\mathcal E\). Two applications concerning linear and multilinear algebra are given. The first one deals with the ``eigendiscriminant locus'', the locus of tensors whose eigenschemes are singular. Eigenvectors of tensors were introduced by \textit{L.-H. Lim} [``Singular values and eigenvalues of tensors: a variational approach'', in: Proceedings of the IEEE International Workshop on Computational Advances in Multisensor Adaptive Processing 1, 129--132 (2005)] and \textit{L. Qi} [J. Symb. Comput. 40, No. 6, 1302--1324 (2005; Zbl 1125.15014)] independently and have been studied in numerical multilinear algebra. The conditions for a non-zero vector to be an eigenvector of a tensor can be described by means of the ideal generated by the \(2 \times 2\) minors of a \(n \times 2\) polynomial matrix. The eigenscheme of the tensor is the scheme defined by this determinantal ideal. Then the eigendiscriminant locus is the locus of tensors whose eigenschemes either are nonreduced or have a positive dimensional component. The author relates the eigenscheme of an \(n\)-dimensional tensor to the zero scheme of a global section of a certain twist \(\mathcal E\) of a Steiner bundle. This allows to relate the eigendiscriminant locus to \(\Delta(\mathcal E)\), deriving properties of the former from those of the latter. The second application concerns the discriminant locus \(\Delta\) of the generic graded matrix of type \((n+r-1)\times (r+1)\), i.e. the locus parameterizing graded matrices of this type with entries homogeneous polynomials in \(k[x_0, \dots ,x_n]\), whose \(r\)-th determinantal schemes either are nonreduced or have positive dimensional components. Even in this context the connection with Steiner bundles is crucial since one can relate \(\Delta\) to the discriminant hypersurface of a Steiner bundle. Thus \(\Delta\) turns out to be a hypersurface, and the formula expressing its degree, which is complicated in general, becomes very simple when the entries of the graded matrix are linear.
scientific article; zbMATH DE number 7376803
scientific article