## Abstract

We study the Hilbert series of two families of ideals generated by powers of linear forms in K [x_{1}, x_{2}, x_{3}]. Using the results of Emsalem-Iarrobino, we formulate this as a question about fatpoints in P^{2}. This is equivalent to studying the dimension of a linear system on a blow up of P^{2}. We determine the classes of the negative curves, then apply an algorithm of Harbourne to reduce to an effective, nef divisor. Combining Harbourne's results on rational surfaces with K^{2} > 0 and Riemann-Roch yields a formula for the Hilbert series. For one family of ideals, this proves the n = 3 case of a conjecture posed by Postnikov and Shapiro "as a challenge to the commutative algebra community" (after this proof was communicated to them, they found a combinatorial proof for all n). For the second family of ideals, it yields a formula which Postnikov and Shapiro were unable to obtain via combinatorial methods. We conjecture a formula for the minimal free resolution of one family of ideals, and show that a member of the second family of ideals provides a counterexample to a conjecture made by Postnikov and Shapiro in [20].

Original language | English (US) |
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Pages (from-to) | 697-713 |

Number of pages | 17 |

Journal | Mathematical Research Letters |

Volume | 11 |

Issue number | 5-6 |

DOIs | |

State | Published - 2004 |

## ASJC Scopus subject areas

- Mathematics(all)