The variety of orthogonal frames

An orthogonal n-frame in a quadratic vector space of dimension d is an ordered set of n pairwise orthogonal vectors. The set of all orthogonal n-frames is an algebraic variety V(d,n). We investigate the variety V(d,n) as well as the quadratic ideal I(d,n) generated by the orthogonality relations, which cuts out V(d,n). We determine the irreducible components, classify when I(d,n) is a complete intersection or a prime ideal, and classify when the variety V(d,n) is normal or factorial. We give many applications to the theory of Lovasz-Saks-Schrijver ideals of simple graphs, and solve problems proposed by Aldo Conca and Volkmar Welker. If time permits, we will discuss some interesting open problems.

This is a joint work with Laura Casabella.