Coherent systems of genus 0

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 $\alpha$. 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 $\dim V=1$ or 2, we determine these conditions precisely. We also obtain partial results in some other cases.