Toric Ideals Generated by Quadratic Binomials

A combinatorial criterion for the toric ideal arising from a finite graph to be generated by quadratic binomials is studied. Such a criterion guarantees that every Koszul algebra generated by squarefree quadratic monomials is normal. We present an example of a normal non-Koszul squarefree semigroup ring whose toric ideal is generated by quadratic binomials as well as an example of a non-normal Koszul squarefree semigroup ring whose toric ideal possesses no quadratic Gröbner basis. In addition, all the affine semigroup rings which are generated by squarefree quadratic monomials and which have 2-linear resolutions will be classified. Moreover, it is shown that the toric ideal of a normal affine semigroup ring generated by quadratic monomials is generated by quadratic binomials if its underlying polytope is simple.