Lovász–Saks–Schrijver ideals and coordinate sections of determinantal varieties

Abstract

Motivated by questions in algebra and combinatorics we study two ideals associated to a simple graph $G$:

• the Lovász-Saks-Schrijver ideal defining the $d$-dimensional orthogonal representations of the graph complementary to $G$, and

• the determinantal ideal of the $\left(d+1\right)$-minors of a generic symmetric matrix with $0$ in positions prescribed by the graph $G$.

In characteristic $0$ these two ideals turn out to be closely related and algebraic properties such as being radical, prime or a complete intersection transfer from the Lovász–Saks–Schrijver ideal to the determinantal ideal. For Lovász–Saks–Schrijver ideals we link these properties to combinatorial properties of $G$ and show that they always hold for $d$ large enough. For specific classes of graphs, such a forests, we can give a complete picture and classify the radical, prime and complete intersection Lovász–Saks–Schrijver ideals.