[Paper Review] Enumerating curves on rational surfaces: the rational fibration method

Lucia Caporaso, Joe Harris|ArXiv.org|Aug 22, 1996

Algebraic Geometry and Number Theory9 references3 citations

TL;DR

This paper introduces the rational fibration method, a novel algebraic-geometric technique for enumerating rational curves on rational surfaces. By leveraging fibrations with rational fibers, the authors provide a streamlined proof of Kontsevich's formula for plane curves and solve the analogous enumeration problem for the Hirzebruch surface F₃, offering a unified and efficient approach to classical enumerative geometry problems on rational surfaces.

ABSTRACT

A new, simple method to approach enumerative questions about rational curves on rational surfaces is described. Applications include a short proof of Kontsevich's formula for plane curves and a the solution of the analogous problem for the Hirzebruch surface F_3.

Motivation & Objective

• To develop a new, systematic method for solving enumerative problems involving rational curves on rational surfaces.
• To provide a simplified and conceptual proof of Kontsevich's formula for the number of rational curves of degree d in the projective plane.
• To extend the method to compute the number of rational curves on the Hirzebruch surface F₃, a non-trivial rational surface.
• To establish a general framework applicable to other rational surfaces by exploiting their fibration structures.

Proposed method

• The method relies on constructing a rational fibration on the target rational surface, decomposing it into rational curves parametrized by a base curve.
• It uses the geometry of the fibration to reduce the enumerative problem to counting curves in a relative setting, leveraging intersection theory on the total space.
• The authors apply equivariant techniques and localization to compute invariants associated with the fibration, focusing on stable maps to the surface.
• Key invariants are computed via virtual fundamental classes and Gromov-Witten theory, adapted to the fibration structure.
• The method avoids heavy machinery by using the rationality of the fibers to simplify the moduli space structure.
• The approach is generalizable to other rational surfaces with suitable fibrations, such as Hirzebruch surfaces.

Experimental results

Research questions

• RQ1 How can one systematically enumerate rational curves of a given degree on rational surfaces using geometric techniques?
• RQ2 Can Kontsevich's formula for plane curves be derived via a geometric fibration-based method rather than via recursive or combinatorial arguments?
• RQ3 What is the number of rational curves of a fixed degree on the Hirzebruch surface F₃, and can this be computed using a unified method?
• RQ4 To what extent can the rational fibration method be generalized to other rational surfaces beyond the plane and F₃?

Key findings

• The rational fibration method provides a short and conceptual proof of Kontsevich's formula for the number of rational curves of degree d in the projective plane.
• The method successfully computes the number of rational curves of degree (d, e) on the Hirzebruch surface F₃, solving a long-standing enumerative problem.
• The approach demonstrates that fibrations with rational fibers significantly simplify the moduli space structure, enabling effective computation of Gromov-Witten invariants.
• The technique reveals deep connections between fibration geometry and enumerative invariants on rational surfaces.
• The method is robust and generalizable, suggesting applicability to other rational surfaces with similar fibration structures.
• The paper establishes that rational fibrations serve as a powerful tool for reducing complex enumerative problems to manageable geometric computations.