Ben Howard

Equations for the moduli space of n labelled points on the projective line

Abstract: A Geometric Invariant Theory quotient of the space of n-tuples of points on the projective line modulo automorphisms of the line is given by positive integer weights w_1,...,w_n where w_i is the weight of the i-th point. This quotient may be regarded as a compactification of the space of distinct n-tuples, where points may collide so long as no more than half the total weight lies at any given point.
The moduli space is Proj of a graded ring R(w), w = (w_1,...,w_n). In the case that n is even and each w_i=1, a theorem of A. B. Kempe (1894) is that R(w) is generated in degree one. However it is easy to prove from Kempe's theorem that R(w) is generated in degree one whenever the total weight w_1 + ... + w_n is even. (If the total weight is odd, then the odd degree pieces of R(w) are all zero, and the even degree subring is isomorphic to R(2w), so then R(w) is generated in degree two.)
By picking an appropriate Groebner degeneration of R(w) to a toric ring, we show that the ideal of relations is generated in degree four and less. If all the points have even weight then the ideal is generated by quadrics of a special sort. (Joint work with J. Millson, A. Snowden, and R. Vakil)