[Paper Review] Welschinger invariant and enumeration of real plane rational curves

Ilia Itenberg, Viatcheslav Kharlamov+1 more|ArXiv.org|Mar 30, 2003

Polynomial and algebraic computation9 references4 citations

TL;DR

This paper establishes a lower bound on the number of real rational curves of degree $d$ passing through $3d-1$ generic real points in the real projective plane using Welschinger's invariant and Mikhalkin's tropical geometry approach. It proves that at least $d!/2$ such real rational curves exist, confirming the existence of real solutions and showing that asymptotically at least one-third of complex rational curves are real in the logarithmic scale.

ABSTRACT

Welschinger's invariant bounds from below the number of real rational curves through a given generic collection of real points in the real projective plane. We estimate this invariant using Mikhalkin's approach which deals with a corresponding count of tropical curves. In particular, our estimate implies that, for any positive integer $d$, there exists a real rational curve of degree $d$ through any collection of $3d-1$ real points in the projective plane, and, moreover, asymptotically in the logarithmic scale at least one third of the complex plane rational curves through a generic point collection are real. We also obtain similar results for curves on other toric Del Pezzo surfaces.

Motivation & Objective

• To establish a non-trivial lower bound on the number of real rational curves passing through a generic configuration of $3d-1$ real points in $\mathbb{R}P^2$.
• To resolve the long-standing open question of whether such real rational curves always exist for any degree $d$.
• To extend the Welschinger invariant to other toric Del Pezzo surfaces and derive analogous lower bounds for rational curves in their linear systems.
• To demonstrate that the Welschinger invariant provides a meaningful lower bound by showing it grows as $d!/2$, implying a positive proportion of complex curves are real asymptotically.

Proposed method

• Uses Welschinger's invariant, which assigns $\pm1$ weights to real rational curves to produce a topological invariant independent of point configuration.
• Applies Mikhalkin's tropical geometry method to count tropical curves, translating the real enumerative problem into a combinatorial count on tropical surfaces.
• Relies on the fact that for Del Pezzo surfaces, Gromov-Witten invariants count irreducible rational curves, enabling the use of Welschinger invariants in this context.
• Establishes a correspondence between real rational curves in algebraic surfaces and their tropical counterparts via degeneration techniques.
• Employs Bézout’s theorem and ramification formulas to analyze the behavior of curves near singularities and base points in pencils.
• Uses the variation formula for ramification order to analyze bifurcations in the real part of the curve space, linking topological changes to changes in the Welschinger invariant.

Experimental results

Research questions

• RQ1 Does there always exist at least one real rational curve of degree $d$ passing through any generic collection of $3d-1$ real points in $\mathbb{R}P^2$?
• RQ2 How non-trivial is the Welschinger invariant as a lower bound for the number of real rational curves?
• RQ3 Can the Welschinger invariant be effectively estimated using tropical geometry methods for Del Pezzo surfaces?
• RQ4 What proportion of complex rational curves through $3d-1$ generic points are real, asymptotically in the logarithmic scale?
• RQ5 Does the Welschinger invariant generalize to curves of positive genus, such as elliptic curves?

Key findings

• For any $d \geq 1$, there exist at least $d!/2$ real rational curves of degree $d$ through any $3d-1$ generic real points in $\mathbb{R}P^2$, confirming the existence of real solutions.
• The logarithmic asymptotic of the lower bound $d!/2$ is $\frac{1}{3} \log N^{\mathbb{C}}_{\mathbb{P}^2,d}$, meaning at least one-third of complex rational curves are real in the logarithmic scale.
• Explicit values of the Welschinger invariant are computed: $W_4 = 240$ and $W_5 = 18264$, confirming the growth rate.
• The lower bound extends to other toric Del Pezzo surfaces: $Q = \mathbb{C}P^1 \times \mathbb{C}P^1$ and $P_k$ (the projective plane with $k$ points blown up), with explicit formulas depending on the divisor class.
• The Welschinger invariant does not generalize directly to elliptic curves, as bifurcations in the real part can cause the invariant to vary with configuration.
• For nodal curves with one node, the Welschinger number is not invariant when $d \geq 4$, as it depends on the number of conjugate imaginary base points in a pencil.