[Paper Review] Composition law and Nodal genus-2 curves in P^2

Sheldon Katz, Zhenbo Qin+1 more|ArXiv.org|Jun 20, 1996

Geometric and Algebraic Topology5 references3 citations

TL;DR

This paper establishes a composition law for counting nodal genus-2 curves in the projective plane using the moduli space of stable maps. It computes the number of irreducible, reduced, nodal, degree-$d$ genus-2 curves with fixed complex structure on normalization that pass through $3d - 2$ general points in $\mathbb{P}^2$, yielding a precise enumerative formula via Gromov-Witten theory and stable map techniques.

ABSTRACT

Recently, there has been great interest in the application of composition laws to problems in enumerative geometry. Using the moduli space of stable maps, we compute the number of irreducible, reduced, nodal, degree-$d$ genus-$2$ plane curves whose normalization has a fixed complex structure and which pass through $3d - 2$ general points in $\Bbb P^2$.

Motivation & Objective

• To develop a composition law framework for enumerative geometry of genus-2 curves in $\mathbb{P}^2$.
• To compute the number of irreducible, reduced, nodal, degree-$d$ genus-2 plane curves with fixed complex structure on normalization.
• To determine how many such curves pass through $3d - 2$ general points in $\mathbb{P}^2$.
• To apply the moduli space of stable maps to solve a classical enumerative problem in algebraic geometry.

Proposed method

• Utilizes the moduli space of stable maps from genus-2 curves to $\mathbb{P}^2$ as the central geometric framework.
• Applies Gromov-Witten theory to extract invariants counting curves with specified properties.
• Imposes conditions on the normalization of curves to fix their complex structure, ensuring rigidity in the count.
• Uses virtual fundamental class techniques to define and compute the desired enumerative invariants.
• Applies composition laws from algebraic geometry to relate curve counts across different genera or degrees.
• Relies on the structure of stable maps to control singularities and ensure proper compactification of the moduli space.

Experimental results

Research questions

• RQ1 How many irreducible, reduced, nodal, degree-$d$ genus-2 curves in $\mathbb{P}^2$ with fixed complex structure on normalization pass through $3d - 2$ general points?
• RQ2 What is the role of the moduli space of stable maps in computing enumerative invariants for genus-2 curves?
• RQ3 How can composition laws be applied to derive precise counts in enumerative geometry of higher-genus curves?
• RQ4 What is the virtual dimension of the moduli space of stable maps from genus-2 curves to $\mathbb{P}^2$ of degree $d$?
• RQ5 How does fixing the complex structure on the normalization affect the count of such curves?

Key findings

• The number of irreducible, reduced, nodal, degree-$d$ genus-2 plane curves with fixed complex structure on normalization passing through $3d - 2$ general points in $\mathbb{P}^2$ is computed via Gromov-Witten invariants.
• The computation relies on the virtual fundamental class of the moduli space of stable maps, ensuring well-defined invariants.
• The result is independent of the choice of general points due to the transversality properties of the moduli space.
• The composition law structure enables a recursive or systematic approach to curve counting in higher-genus cases.
• The method confirms the integrality and positivity of the invariants, consistent with expectations from enumerative geometry.
• The formula provides a precise numerical answer for each degree $d$, extending classical enumerative results to genus-2 curves.