Gorenstein Braid Cones and Crepant Resolutions.
Saved in:
| Title: | Gorenstein Braid Cones and Crepant Resolutions. |
|---|---|
| Authors: | Hallam, Joshua1 (AUTHOR), Machacek, John2 (AUTHOR) johnmach@uoregon.edu |
| Source: | Discrete & Computational Geometry. Apr2024, Vol. 71 Issue 3, p1021-1056. 36p. |
| Subjects: | Toric varieties, Möbius function, Partially ordered sets |
| Abstract: | To any poset P, we associate a convex cone called a braid cone. We also associate a fan and study the toric varieties the cone and fan define. The fan always defines a smooth toric variety X P , while the toric variety U P of the cone may be singular. We show that X P ⤏ U P is a crepant resolution of singularities if and only if P is bounded. Next, we aim to determine when U P is Q -Gorenstein or Gorenstein. We prove that whether or not U P is Q -Gorenstein or Gorenstein depends only on the biconnected components of the Hasse diagram of P. In the case that P has a minimum or maximum element, we show that the Gorenstein property of U P is completely determined by the Möbius function of P. We also provide a recursive method that determines if U P is Q -Gorenstein or Gorenstein in this case. We conjecture that U P is Gorenstein if and only if it is Q -Gorenstein. We verify this conjecture for posets of length 1 and also for posets with a minimum or maximum element. [ABSTRACT FROM AUTHOR] |
| Copyright of Discrete & Computational Geometry is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
|
Full text is not displayed to guests.
Login for full access.
|
|
| Abstract: | To any poset P, we associate a convex cone called a braid cone. We also associate a fan and study the toric varieties the cone and fan define. The fan always defines a smooth toric variety X P , while the toric variety U P of the cone may be singular. We show that X P ⤏ U P is a crepant resolution of singularities if and only if P is bounded. Next, we aim to determine when U P is Q -Gorenstein or Gorenstein. We prove that whether or not U P is Q -Gorenstein or Gorenstein depends only on the biconnected components of the Hasse diagram of P. In the case that P has a minimum or maximum element, we show that the Gorenstein property of U P is completely determined by the Möbius function of P. We also provide a recursive method that determines if U P is Q -Gorenstein or Gorenstein in this case. We conjecture that U P is Gorenstein if and only if it is Q -Gorenstein. We verify this conjecture for posets of length 1 and also for posets with a minimum or maximum element. [ABSTRACT FROM AUTHOR] |
|---|---|
| ISSN: | 01795376 |
| DOI: | 10.1007/s00454-023-00589-8 |
