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Toric Ideals Generated by Quadratic Binomials

✍ Scribed by Hidefumi Ohsugi; Takayuki Hibi

Publisher
Elsevier Science
Year
1999
Tongue
English
Weight
157 KB
Volume
218
Category
Article
ISSN
0021-8693

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✦ Synopsis

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.

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