Equations and syzygies of some Kalman varieties (Q2846840)

Let \(V\) be a \(\mathbb{K}\)-vector space, where \(\mathbb{K}\) an arbitrary field and let \(\dim V=n\). Let also \(L\) be a subspace of \(V\) with \(\dim L=d\). For \(1\leq s\leq d\) we define the \textit{Kalman variety} over the vector space \(V\) as: NEWLINE\[NEWLINE\mathcal{K}_{s,d,n}=\{\varphi \in \text{End} (V)\;| \;\text{there exists}\;U\subset L\;\text{such that}\;\dim U=s\;\text{and}\;\varphi(U)\subseteq U\}NEWLINE\]NEWLINE where \(\text{End} (V)\) is the space of linear operators on \(V\). Finally with \(\mathcal{O}_{s,d,n}\) we denote the coordinate ring of \(\mathcal{K}_{s,d,n}\) and let \(\mathcal{\tilde{O}}_{s,d,n}\) be its normalization.NEWLINENEWLINEKalman varieties were defined by \textit{R. E. Kalman} in [Bol. Soc. Mat. Mex., II. Ser. 5, 102--119 (1960); erratum ibid. 5, 299 (1960; Zbl 0112.06303)] and were studied more especially by \textit{G. Ottaviani} and \textit{B. Sturmfels} [Proc. Am. Math. Soc. 141, No. 4, 1219--1232 (2013; Zbl 1272.15010)]. Ottaviani and Sturmfels described the Kalman variety for a prime ideal in the case \(\mathcal{K}_{s,2,n}\) and conjectured the number of minimal equations which is needed when \(d=3\), in other words the case \(\mathcal{K}_{s,3,n}\).NEWLINENEWLINEIn this paper the author proves in his main result the conjecture which is described above. Furthermore he describes the minimal free resolution in the case \(\mathcal{K}_{s,2,n}\). In order to achieve this, the author uses as main tools some results on the ring \(\mathcal{O}_{s,d,n}\) and mainly an exact sequence that involves the normalization \(\mathcal{\tilde{O}}_{s,d,n}\) in the case that \(d\leq3\). In the general case of any dimension \(d\) of \(L\), the author conjectured the existence of this exact sequence.