On the Dimension of the Locus of Determinantal Hypersurfaces

Reichstein, Zinovy ; Vistoli, Angelo

Canadian mathematical bulletin, Tome 60 (2017) no. 3, pp. 613-630

Résumé

The characteristic polynomial ${{P}_{A}}({{x}_{0}},...,{{x}_{r}})$ of an $r$ -tuple $A\,:=({{A}_{1}},...,{{A}_{r}})$ of $n\times n$ -matrices is defined as 1 $${{P}_{A}}({{x}_{0}},...,{{x}_{r}}):=\det ({{x}_{0}}I+{{x}_{1}}{{A}_{1}}+\ldots +{{x}_{r}}{{A}_{r}}).$$ We show that if $r\,\,3$ and $A\,:=({{A}_{1}},...,{{A}_{r}})$ is an $r$ -tuple of $n\times n$ -matrices in general position, then up to conjugacy, there are only finitely many $r$ -tuples $A'\,:=(A_{1}^{'},...,A_{r}^{'})$ such that ${{p}_{A}}={{p}_{A'}}$ . Equivalently, the locus of determinantal hypersurfaces of degree $n$ in ${{\text{P}}^{r}}$ is irreducible of dimension $(r-1){{n}^{2}}+1$ .

Détail

DOI : 10.4153/CMB-2016-070-8

Mots-clés : 14M12, 15A22, 05A10, determinantal hypersurface, matrix invariant, q-binomial cofficient