[Paper Review] Geometries enumeratives complexe, reelle et tropicale

Erwan Brugallé|arXiv (Cornell University)|Jul 31, 2008

Mathematics and Applications3 citations

TL;DR

This paper introduces tropical geometry as a powerful tool for solving complex and real enumerative geometry problems, demonstrating that tropical curves can be used to compute classical invariants like Gromov-Witten and Welschinger invariants. By leveraging Mikhalkin's correspondence theorems and floor diagrams, it provides a combinatorial framework to enumerate rational curves in the complex and real projective plane, yielding exact counts independent of point configuration and establishing lower bounds for real curves via Welschinger invariants.

ABSTRACT

This text is an introduction to algebraic enumerative geometry and to applications of tropical geometry to classical geometry, based on a course given during the X-UPS mathematical days, 2008 May 14th and 15th. The aim of this text is to be understandable by a first year master student.

Motivation & Objective

• To provide an accessible introduction to enumerative geometry for first-year master's students.
• To explain how tropical geometry simplifies the enumeration of complex and real algebraic curves.
• To demonstrate the use of floor diagrams and tropical curves for computing invariants such as Welschinger and Gromov-Witten numbers.
• To show that tropical methods yield configuration-independent counts, even in the real case via signed invariants.
• To present a combinatorial algorithm for computing Welschinger invariants using tropical correspondence theorems.

Proposed method

• Utilizes Mikhalkin’s correspondence theorem to equate counts of complex or real curves with counts of tropical curves.
• Employs floor diagrams as a combinatorial tool to enumerate rational curves in the tropical setting.
• Applies recursive formulas and generating functions to compute invariants such as W(d) and N(d,0).
• Uses logarithmic asymptotics to derive lower bounds on the number of real rational curves through 3d−1 points.
• Introduces the concept of 'staircase decomposition' of tropical curves to reduce enumeration to diagram counting.
• Applies signed counts (Welschinger invariants) to ensure invariance under point configuration in real enumerative geometry.

Experimental results

Research questions

• RQ1 How can tropical geometry be used to compute the number of complex rational curves passing through a given number of points?
• RQ2 Why does the number of real rational curves depend on the configuration of points, and how can this be overcome?
• RQ3 Can Welschinger invariants provide a configuration-independent lower bound on the number of real rational curves?
• RQ4 How do floor diagrams simplify the enumeration of tropical curves and relate to classical invariants?
• RQ5 What is the asymptotic behavior of the number of real rational curves through 3d−1 points in the plane?

Key findings

• The number of complex rational curves of degree d passing through 3d−1 general points in the complex projective plane is given by Kontsevich’s formula, and tropical geometry provides a combinatorial proof via floor diagrams.
• For real rational curves, the Welschinger invariant W(d) provides a signed count that is independent of the configuration of 3d−1 points, ensuring a lower bound on the number of real curves.
• The paper proves that W(d) ≥ (3d−4)! − d ln 48 − 5∑(i=1 to l−1) 2^i ln(3n/2^i + 1), which grows asymptotically as 3d ln d, showing that at least exponentially many real rational curves exist.
• The use of tropical curves and their decomposition into floor diagrams allows for the exact computation of Welschinger invariants, as demonstrated by Mikhalkin’s algorithm.
• The correspondence theorem ensures that tropical counts of curves in R² match the Gromov-Witten invariants of CP², validating the method.
• The paper establishes that the Welschinger invariant is logarithmically equivalent to the Gromov-Witten invariant, confirming deep connections between real and complex enumerative geometry.