Lotuses as computational architectures

TL;DR

The paper presents lotuses as a unifying combinatorial framework for embedded plane curve resolutions, encoding Enriques diagrams, dual graphs, and resolution data into a single two-dimensional simplicial complex. It develops active constellations of crosses and their lotus to compute key invariants—log-discrepancies $\lambda$, orders of vanishing $\mathrm{ord}_E(L)$, multiplicities $e_P(A)$, intersection numbers, and the Eggers-Wall tree $\Theta_L(A)$—and demonstrates how to derive these invariants directly from the lotus. It further connects lotuses to Newton lotuses, continued fractions, and EW trees, providing algorithms to construct EW trees from lotus data and to reconstruct lotuses from EW trees, including methods to compute the semigroup generators, delta invariant $\delta(A)$, and Milnor number $\mu(A)$. The work extends these constructions to positive characteristic, introduces NP-based EW trees when Newton-Puiseux roots exist, and discusses the implications for deformations and experiments in plane curve singularity theory, offering a versatile computational toolkit with potential broader impact in singularity theory and toric/resolution geometry.

Abstract

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.

Figures (45)

• Figure 1: An example of lotus
• Figure 2: The process of minimal embedded resolution of a cusp singularity $A$
• Figure 3: The blowup process leading to the model of the active constellation $\{O_0, \dots, O_3\}$ in which $O_3$ is inactive
• Figure 4: The blowup process leading to the model of the active constellation $\{O_0, \dots, O_3\}$ in which $O_3$ is active
• Figure 5: Extension of the active constellation of Figure \ref{['fig:assconstcusp']} into an active constellation of crosses

Theorems & Definitions (157)

• Definition 2.1
• Theorem 2.2
• Definition 2.3
• Proposition 2.4
• proof
• Definition 3.1
• Definition 3.2
• Theorem 3.3
• Example 3.4
• Definition 4.1