#### Vol. 13, No. 2, 2019

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

### Aldo Conca and Volkmar Welker

Vol. 13 (2019), No. 2, 455–484
##### 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.

##### Keywords
determinantal rings, complete intersections, ideals associated to graphs, Gröbner bases.
##### Mathematical Subject Classification 2010
Primary: 05E40
Secondary: 05C62, 13P10
##### Milestones
Received: 8 February 2018
Revised: 3 November 2018
Accepted: 24 December 2018
Published: 2 March 2019
##### Authors
 Aldo Conca Dipartimento di Matematica Università di Genova 16146 Genova Italy Volkmar Welker Fachbereich Mathematik und Informatik Philipps-Universität Marburg 35032 Marburg Germany