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On the pole placement problem for linear systems of arbitrary genus. (English) Zbl 0956.14023

V. G. Lomadze introduced [Acta Appl. Math. 34, No. 3, 305-312 (1994; Zbl 0798.93027)] linear systems associated to divisors and vector bundles on \(X\) of genus \(g>0\). We consider the pole placement problem in the setting and with the notation of this cited paper. Let \(F\) be a rank \(m\) effective vector bundle on an arbitrary complete, smooth, and irreducible algebraic curve \(X\) of genus \(g>0\); the pole placement problem (or PPP for short) asks if there is always a completely reachable linear system \(s\) with \(m\) inputs, \(F\) as associated Martin-Hermann vector bundle and the determinant \(\text{ch}(F)\) of \(F\) as characteristic divisor \(c(s)\).
In this paper for every integer \(m\geq 2\) and for every such curve \(X\) we will give a negative solution to PPP for the case of \(m\) inputs. Recall that for genus 0 PPP has always a solution. For the case of one input (i.e. the rank \(m=1\) case) V. G. Lomadze (loc. cit.; §7, theorem 3) has shown that PPP has always a solution. Indeed we will show that for \(g>0\) and \(m\geq 2\) the non solubility of PPP is rather the norm, not an exception. We will use heavily the notions introduced in the paper cited above.

MSC:

14H60 Vector bundles on curves and their moduli
93B27 Geometric methods
14C20 Divisors, linear systems, invertible sheaves
93B55 Pole and zero placement problems
93C05 Linear systems in control theory

Citations:

Zbl 0798.93027
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