### Abstract

We determine the defining equations of the Rees algebra of an ideal I in the case where I is a square-free monomial ideal such that each connected component of the line graph of the hypergraph corresponding to I has at most 5 vertices. Moreover, we show in this case that the non-linear equations arise from even closed walks of the line graph, and we also give a description of the defining ideal of the toric ring when I is generated by square-free monomials of the same degree. Furthermore, we provide a new class of ideals of linear type. We show that when I is a square-free monomial ideal with any number of generators and the line graph of the hypergraph corresponding to I is the graph of a disjoint union of trees and graphs with a unique odd cycle, then I is an ideal of linear type.

Original language | English (US) |
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Pages (from-to) | 25-54 |

Number of pages | 30 |

Journal | Journal of Commutative Algebra |

Volume | 7 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2015 |

### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

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## Cite this

*Journal of Commutative Algebra*,

*7*(1), 25-54. https://doi.org/10.1216/JCA-2015-7-1-25