Seminar Announcement
Speaker:
Professor Benjamin Howard
University of Michigan
Title: A representation-theoretic study of the coordinate ring of eight ordered points on the line.
Abstract:
(co-authors: John Millson, Andrew Snowden, and Ravi Vakil)
The projective coordinate ring of the moduli space M_8 of eight ordered points on the line is a quotient of Sym(V) by an ideal I, where V is an irreducible fourteen-dimensional representation of the symmetric group S_8.
We show that there is a unique cubic hypersurface S in projective space P(V) which is stable under S_8, whose equation s is skew-invariant, and that the singular locus of S is the modular fivefold M_8. In characteristic zero, we prove (without a computer) via commutative algebra and representation theory, that the (fourteen) partial derivatives of s generate the ideal I, and we find the graded Betti numbers of a minimal free resolution.
Using a computer we can strengthen this: over the integers Z, the cubic s and its partials generate I, and using this computer-aided fact, we find the graded Betti numbers over a field of arbitrary characteristic.
The existence of such a cubic was predicted by Igor Dolgachev. We will use this result in an upcoming paper "The ring of projective invariants of n ordered points on the line", as part of the proof (in fact as the base case of a proof by induction) of the main theorem of that paper, which is that the ideal of relations among projective invariants of n weighted points on the line is generated by quadrics unless there are 6 points, each of weight one. This project completes the program of the paper "The equations for the moduli space of n points on the line", to appear in Duke Math Journal.