Coherent systems of genus 0

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In this paper we begin the classification of coherent systems (E,V) on the projective line which are stable with respect to some value of a parameter α. In particular we show that the moduli spaces, if non-empty, are always smooth and irreducible of the expected dimension. We obtain necessary conditions for non-emptiness and, when V=1 or 2, we determine these conditions precisely. We also obtain partial results in some other cases.