On the algebro-geometric structure of the moduli space of completely reachable systems (Q1071086)

Algebro-geometric properties are studied for the moduli space of completely reachable pairs arising in the algebraic theory of dynamical systems. Let k be an algebraically closed field of characteristic zero, \(M_{n,m}\) be the set of \(n\times m\) matrices with coefficients from k. Let \((L,N)\in M_{n,n}\times M_{n,m}\). The pair is called completely reachable if its stabilizer is trivial in case of the action of GL(n) transforming (L,N) in \((gLg^{-1},gN)\). Let us consider the set \(V_{n,m}\) of all completely reachable pairs. We define the morphism \(\phi\) by the following diagram: \(V_{n,m}\hookrightarrow M_{n,n}\times M_{n,m}\to M_{n,n}\to^{\alpha}k^ n;\quad V_{n,m}\to^{\pi}M_{n,m}\to^{\phi}k^ n.\) Here \(\pi\) is the geometric factorization according to the action of GL(n), \(\alpha (L)=(C_ 1(L),...,C_ n(L))\), where \(C_ i(L)\) are coefficients of the characteristic polynomials det(T\(\cdot id-L)\). The structure of the fibres \(Z_ a\) of the morphism \(\phi\) is studied. Main results: The fibre \(Z_ a\) is a normal projective Cohen-Macaulay variety with rational singularities. The set of non-singular points of the fibre \(Z_ a\) is naturally isomorphic to a product of vector bundles over \({\mathbb{P}}^{m-1}\).