Parametrizing Rings of Rank 6

Abstract

We give various methods of parametrizing sextic rings (rings of rank 6) seeking to provide limited generalizations of Bhargava's Higher Composition Laws.Most of the parametrization methods take form of constructing sextic rings from certain algebraic forms. For A∈\Z3⊗\Z3⊗\Z5 a 3×3×5 form, we construct a sextic ring RA whose multiplication coefficients are given as polynomials in entries of A. The ring RA is constructed from hypercohomology of a minimal free resolution of six points in \pp2 determined by certain equations in A, and it can be realized as subring of the ring of global functions on those six points. Similarly, we construct a sextic ring from an element of \Z2⊗\Z2⊗\Z2⊗\Z5 from hypercohomology of a minimal free resolution of six points in \pp1×\pp1 determined by certain equations. The parametrization method in the Chapter 5 takes opposite approach -- it constructs a Segre cubic threefold from a sextic ring with nonzero discriminant. This gives a one-to-one correspondence between primitive sextic nondegenerate rings and Segre cubics with integral coefficients with gcd of its coefficients 1 up to \GL5(\Z)-equivalence. This method allows us to give a geometric interpretation for the Higher Composition Law for quintic rings. For a form A∈\Z4⊗∧2\Z5 which parametrizes a pair (R,S) of a quintic ring R and its sextic resolvent ring S in the quintic Higher Composition Law, one can construct a Segre cubic from A whose corresponding sextic ring becomes a subring of S⊗\Q.