[Paper Review] Tropicalization of Del Pezzo Surfaces

Qingchun Ren, Kristin Shaw+1 more|arXiv (Cornell University)|Feb 23, 2014

Polynomial and algebraic computation20 references3 citations

TL;DR

This paper studies the tropicalization of very affine del Pezzo surfaces—specifically, cubic surfaces (degree 3), degree 4, and degree 5 surfaces—obtained by removing their 27, 16, and 10 exceptional (-1)-curves, respectively. Using three complementary methods—Cox ideals, moduli fan structures, and tropical modifications—it shows that the tropicalizations are characterized by metric trees at infinity, glued according to the Petersen (degree 5), Clebsch (degree 4), and Schläfli (degree 3) graphs, with two distinct generic types for cubic surfaces, differing in bounded cell count and tree valence.

ABSTRACT

We determine the tropicalizations of very affine surfaces over a valued field that are obtained from del Pezzo surfaces of degree 5, 4 and 3 by removing their (-1)-curves. On these tropical surfaces, the boundary divisors are represented by trees at infinity. These trees are glued together according to the Petersen, Clebsch and Schläfli graphs, respectively. There are 27 trees on each tropical cubic surface, attached to a bounded complex with up to 73 polygons. The maximal cones in the 4-dimensional moduli fan reveal two generic types of such surfaces.

Motivation & Objective

• To understand the tropical geometry of very affine del Pezzo surfaces by studying the tropicalization of the complement of their (-1)-curves.
• To characterize the tropical surfaces as intrinsic objects in the moduli space of cubic surfaces.
• To reveal how the 27 lines (as (-1)-curves) appear as metric trees at infinity in the tropicalization.
• To establish a unified framework using three distinct computational techniques: Cox ideals, moduli fan structures, and tropical modifications.
• To extend the understanding of tropical models beyond embedded tropical surfaces to intrinsic tropicalizations of open surfaces.

Proposed method

• The authors use the Cox ideal of the del Pezzo surface to derive a system of trinomial relations, which are then tropicalized using the software gfan.
• They construct the moduli space of cubic surfaces as the tropicalization of a very affine variety, identifying it with the 4-dimensional Naruki fan.
• The moduli fan is analyzed via its maximal cones, revealing two W(E6)-orbits corresponding to two generic types of tropical cubic surfaces.
• Tropical modifications are applied to the tropical projective plane to build the tropical del Pezzo surfaces combinatorially, mirroring classical blow-ups at points.
• The construction is validated by showing consistency across all three methods: Cox ideals, moduli fans, and tropical modifications.
• The resulting tropical surfaces are analyzed combinatorially, with bounded cells, rays, flaps, and tree structures explicitly computed.

Experimental results

Research questions

• RQ1 How do the 27 (-1)-curves on a cubic surface manifest in the intrinsic tropicalization of the complement surface?
• RQ2 What are the two generic combinatorial types of tropical cubic surfaces, and how do they differ in their bounded and unbounded structures?
• RQ3 How are the metric trees at infinity on tropical del Pezzo surfaces of degrees 3, 4, and 5 related to the underlying root systems and graph structures such as the Petersen, Clebsch, and Schläfli graphs?
• RQ4 Can the tropicalization of the complement of the 27 lines on a cubic surface be computed consistently via multiple methods—Cox ideals, moduli fans, and tropical modifications?
• RQ5 Do the 270 trinomial relations in the Cox ideal of a cubic surface form a tropical basis, ensuring the correctness of the tropicalization?

Key findings

• There are two generic types of tropical cubic surfaces, differing in the number of bounded cells (73 vs. 72), edges (150 vs. 148), and vertices (78 vs. 77), with the second type having three 4-valent trees.
• The first type has 27 trivalent trees at infinity, while the second has three 4-valent trees and 24 trivalent trees, with all trees having 10 leaves.
• For degree 5 del Pezzo surfaces, the tropicalization is the cone over the Petersen graph, with 16 trivalent trees, 9 bounded cells, and 48 rays.
• For degree 4 del Pezzo surfaces, the tropicalization is contractible and characterized by 16 trivalent trees, each with 5 leaves, forming a structure dual to the Clebsch graph.
• The 27 trees on the tropical cubic surface are isomorphic, with their leaves partitioned into 10 = 4+3+3 via orthogonality to A2 subroot systems in E7\E6.
• The authors conjecture that the 270 trinomial relations in the Cox ideal of a cubic surface form a tropical basis, supporting the correctness of the tropicalization.