TORIC IDEALS AND DIAGONAL 2-MINORS

Abstract. Let G be a simple graph on the vertex set {1,..., n}. An algebraic object attached to G is the ideal PG generated by diagonal 2-minors of an n × n matrix of variables. In this paper we first provide some general results concerning the ideal PG. It is also proved that if G is bipartite, then every initial ideal of PG is generated by squarefree monomials. Furthermore, we completely characterize all graphs G for which PG is the toric ideal associated to a finite simple graph. As a byproduct we obtain classes of toric ideals associated to non-bipartite graphs which have quadratic Gröbner bases. Finally, we provide infor-mation in certain cases about the universal Gröbner basis of PG. 1