[Paper Review] Higher genus curves on toric varieties

Mihai Halic|ArXiv.org|Jan 11, 2001

Algebraic Geometry and Number Theory4 references3 citations

TL;DR

This paper develops a combinatorial method to compute intersection numbers on a smooth compactification of the space of morphisms from a fixed genus-g curve to a smooth, projective toric variety, under the condition that the morphism degree is sufficiently large. Using equivariant cohomology and localization on the Picard torus of the curve, it derives explicit formulas for certain intersection products and establishes vanishing results based on primitive collections of the fan defining the toric variety.

ABSTRACT

Given a smooth and projective curve C and a smooth and projective toric variety X, we first describe a compactification of the space of morphisms from C to X representing a fixed homology class, and after we study the intersection theory on this variety.

Motivation & Objective

• To describe the space of morphisms from a fixed smooth, projective curve of genus g to a smooth, projective toric variety when the morphism degree is sufficiently large.
• To construct a smooth, projective compactification of this space of morphisms that is birational to an irreducible component of the Kontsevich-Manin stable map space.
• To compute intersection numbers on this compactified space using equivariant cohomology and localization techniques.
• To derive explicit formulae for specific intersection products and establish vanishing conditions based on the combinatorics of the toric variety's fan.

Proposed method

• Uses the toric variety's fan structure to define a T-action on C^r and realize X as a quotient Ω/T, with Z_X = C^r ∕ Ω of codimension ≥2.
• Constructs a compactification V of the space Mor_d(C,X) of morphisms of multi-degree d, which is smooth and of expected dimension when d is large.
• Applies equivariant localization on the Picard torus of the curve C to reduce integration over V to integration over fixed loci in a product of Picard varieties.
• Uses the structure of the fan to define theta classes and line bundles Λ_ρ on the compactification, and computes pushforwards via the formula (q_x)_*(∏Λ_ρ^{k_ρ}) using known results on projective bundles.
• Applies the localization formula (Theorem 4.5) to express integrals over V as sums over fixed points x ∈ X^S, with contributions from each x depending on which rays ρ are in σ_n(x).
• Imposes conditions on exponents m_ρ to ensure that terms with r ∈ σ_n(x) vanish, allowing simplification by setting variables to zero, leading to explicit formulae in Proposition 4.7.

Experimental results

Research questions

• RQ1 How can intersection numbers on the space of morphisms from a genus-g curve to a toric variety be computed when the morphism degree is large enough for the space to be smooth and of expected dimension?
• RQ2 What is the structure of the compactification of the space of such morphisms, and how does it relate to the Kontsevich-Manin stable map compactification?
• RQ3 Can explicit formulae be derived for specific intersection products on this compactified space using equivariant localization?
• RQ4 Under what combinatorial conditions on the fan does the integral of a monomial in the Λ_ρ classes vanish?
• RQ5 How do the theta classes of the Picard variety and the fan combinatorics jointly determine the intersection numbers?

Key findings

• For sufficiently large degree, the space of morphisms Mor_d(C,X) is smooth and of expected dimension, allowing for a well-behaved compactification V.
• The integral ∫_V Λ_1^{m_1}⋯Λ_r^{m_r} vanishes if the indices {ρ} with m_ρ ≥ N_ρ + g form a primitive collection of the fan.
• When m_ρ = d_ρ + 1 + a_ρ for ρ=1,…,r−1 and m_r = d_r − ng − (l + a_r − 1) with d_r large enough, the integral simplifies to a sum over fixed points x ∈ X^S with r ∉ σ_n(x), and the contribution is given by (q_x)_* of a product of Λ_ρ^{m_ρ} and correction terms involving θ_ρ^b / b!.
• The formula in Proposition 4.7 expresses the pushforward q_*Λ^{m_ρ} as a sum over fixed points x with r ∉ σ_n(x), involving pushforwards over projective bundles with known degree formulas.
• The vanishing result in Proposition 4.8 shows that if a set J ⊂ Σ(1) is a primitive collection and m_ρ ≥ N_ρ + g for all ρ ∈ J, then the integral vanishes, due to the absence of such x with J ⊂ σ_n(x).
• The entire integration problem on V is reduced to combinatorial data from the fan and the theta classes of the Picard variety, showing that intersection numbers depend only on the toric variety’s combinatorics and the curve’s moduli.