ar X iv : m at h / 05 05 09 6 v 4 [ m at h . A G ] 7 N ov 2 00 5 THE PROJECTIVE INVARIANTS OF ORDERED POINTS ON THE LINE I

In this paper we study the structure of the graded ring of projective invariants of n ordered points on the projective line CP. A theorem of Kempe (1894) [Ke], see also [Howe] and [KR], states that the graded ring R of projective invariants is generated by elements of lowest degree. We give a new proof of this theorem using a toric degeneration. Our main theorem implies that the ideal of relations in the lowest degree elements is generated by quadratic relations if n is odd and by relations of degrees two, three, and four if n is even. We also show that R is not a complete intersection except for n ≤ 4 and n = 6.