## Abstract

We consider the conjecture that all artinian height 4 complete intersections of forms of the same degree d have the Weak Lefschetz Property (WLP). We translate this problem to one of studying the general hyperplane section of a certain smooth curve in P^{3}, and our main tools are the Socle Lemma of Huneke and Ulrich together with a careful liaison argument. Our main results are (i) a proof that the property holds for d = 3, 4 and 5; (ii) a partial result showing maximal rank in a non-trivial but incomplete range, cutting in half the previous unknown range; and (iii) a proof that maximal rank holds in a different range, even without assuming that all the generators have the same degree. We furthermore conjecture that if there were to exist any height 4 complete intersection generated by forms of the same degree and failing WLP then there must exist one (not necessarily the same one) failing by exactly one (in a sense that we make precise). Based on this conjecture we outline an approach to proving WLP for all equigenerated complete intersections in four variables. Finally, we apply our results to the Jacobian ideal of a smooth surface in P^{3}.

Original language | English |
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Pages (from-to) | 1254-1286 |

Number of pages | 33 |

Journal | Transactions of the American Mathematical Society Series B |

Volume | 10 |

DOIs | |

State | Published - 2023 |

### Bibliographical note

Publisher Copyright:©2023 by the author(s) under Creative Commons.

### Funding

Received by the editors December 26, 2022, and, in revised form, April 15, 2023. 2020 Mathematics Subject Classification. Primary 14M10, 13D40; Secondary 14H50, 14M06. The first author was partially supported by the grant VR2013-4545; the second author was partially supported by grants from the Simons Foundation (#208579, #309556, and #839618); the third author was partially supported by the grant PID2019-104844GB-I00; the fourth author was partially supported by grants from the Simons Foundation (#317096 and #636513).

Funders | Funder number |
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Simons Foundation | 636513, 317096, 839618, PID2019-104844GB-I00, 309556, 208579 |

## ASJC Scopus subject areas

- Mathematics (miscellaneous)