Lotuses as computational architectures
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909508170678272 |
|---|---|
| author | Barroso, Evelia R. García Pérez, Pedro D. González Popescu-Pampu, Patrick |
| author_facet | Barroso, Evelia R. García Pérez, Pedro D. González Popescu-Pampu, Patrick |
| contents | Lotuses are certain types of finite contractible simplicial complexes, obtained by identifying vertices of polygons subdivided by diagonals. As we explained in a previous paper, each time one resolves a complex reduced plane curve singularity by a sequence of toroidal modifications with respect to suitable local coordinates, one gets a naturally associated lotus, which allows to unify the classical trees used to encode the combinatorial type of the singularity. In this paper we explain how to associate a lotus to each constellation of crosses, which is a finite constellation of infinitely near points endowed with compatible germs of normal crossings divisors with two components, and how this lotus may be seen as a computational architecture. Namely, if the constellation of crosses is associated to an embedded resolution of a complex reduced plane curve singularity $A$, one may compute progressively as vertex and edge weights on the lotus the log-discrepancies of the exceptional divisors, the orders of vanishing on them of the starting coordinates, the multiplicities of the strict transforms of the branches of $A$, the orders of vanishing of a defining function of $A$, the associated Eggers-Wall tree, the delta invariant and the Milnor number of $A$, etc. We illustrate these computations using three recurrent examples. Finally, we describe the changes to be done when one works in positive characteristic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_17102 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Lotuses as computational architectures Barroso, Evelia R. García Pérez, Pedro D. González Popescu-Pampu, Patrick Algebraic Geometry Combinatorics 14H20 (primary), 14B05, 32S05 Lotuses are certain types of finite contractible simplicial complexes, obtained by identifying vertices of polygons subdivided by diagonals. As we explained in a previous paper, each time one resolves a complex reduced plane curve singularity by a sequence of toroidal modifications with respect to suitable local coordinates, one gets a naturally associated lotus, which allows to unify the classical trees used to encode the combinatorial type of the singularity. In this paper we explain how to associate a lotus to each constellation of crosses, which is a finite constellation of infinitely near points endowed with compatible germs of normal crossings divisors with two components, and how this lotus may be seen as a computational architecture. Namely, if the constellation of crosses is associated to an embedded resolution of a complex reduced plane curve singularity $A$, one may compute progressively as vertex and edge weights on the lotus the log-discrepancies of the exceptional divisors, the orders of vanishing on them of the starting coordinates, the multiplicities of the strict transforms of the branches of $A$, the orders of vanishing of a defining function of $A$, the associated Eggers-Wall tree, the delta invariant and the Milnor number of $A$, etc. We illustrate these computations using three recurrent examples. Finally, we describe the changes to be done when one works in positive characteristic. |
| title | Lotuses as computational architectures |
| topic | Algebraic Geometry Combinatorics 14H20 (primary), 14B05, 32S05 |
| url | https://arxiv.org/abs/2502.17102 |