amari-enumerative

Enumerative geometry for counting geometric configurations.

Overview

amari-enumerative provides tools for enumerative geometry, the mathematical discipline concerned with counting geometric objects satisfying given conditions. The crate implements intersection theory, Schubert calculus, Gromov-Witten invariants, and tropical curve counting.

Features

• Intersection Theory: Chow rings, intersection multiplicities, Bézout's theorem
• Schubert Calculus: Computations on Grassmannians and flag varieties
• Gromov-Witten Theory: Curve counting and quantum cohomology
• Tropical Geometry: Tropical curve counting via correspondence theorems
• Moduli Spaces: Computations on moduli spaces of curves
• GPU Acceleration: Optional GPU support for large computations

Installation

Add to your Cargo.toml:

[dependencies]
amari-enumerative = "0.12"

Feature Flags

[dependencies]
# Default features
amari-enumerative = "0.12"

# With serialization
amari-enumerative = { version = "0.12", features = ["serde"] }

# With GPU acceleration
amari-enumerative = { version = "0.12", features = ["gpu"] }

# With parallel computation
amari-enumerative = { version = "0.12", features = ["parallel"] }

# For WASM targets
amari-enumerative = { version = "0.12", features = ["wasm"] }

Quick Start

use amari_enumerative::{ProjectiveSpace, ChowClass, IntersectionRing};

// Create projective 2-space (P²)
let p2 = ProjectiveSpace::new(2);

// Define two curves by degree
let cubic = ChowClass::hypersurface(3);   // Degree 3 curve
let quartic = ChowClass::hypersurface(4); // Degree 4 curve

// Compute intersection number using Bézout's theorem
let intersection = p2.intersect(&cubic, &quartic);
assert_eq!(intersection.multiplicity(), 12); // 3 × 4 = 12 points

Key Concepts

Bézout's Theorem

Two plane curves of degrees d and e intersect in d·e points (counting multiplicity):

use amari_enumerative::{ProjectiveSpace, ChowClass};

let p2 = ProjectiveSpace::new(2);
let line = ChowClass::hypersurface(1);
let conic = ChowClass::hypersurface(2);

// Line meets conic in 1 × 2 = 2 points
let points = p2.intersect(&line, &conic).multiplicity();
assert_eq!(points, 2);

Schubert Calculus

Count linear subspaces satisfying incidence conditions:

use amari_enumerative::{Grassmannian, SchubertClass};

// Grassmannian Gr(2,4): lines in P³
let gr = Grassmannian::new(2, 4);

// How many lines meet 4 general lines in P³?
let sigma_1 = SchubertClass::sigma(&[1]); // Lines meeting a line
let count = gr.intersect(&[sigma_1; 4]);
assert_eq!(count, 2); // Answer: 2 lines

Gromov-Witten Invariants

Count curves in algebraic varieties:

use amari_enumerative::{GromovWittenInvariant, QuantumCohomology};

// Count rational curves of degree d in P²
let gw = GromovWittenInvariant::new(variety, degree);
let count = gw.compute_with_insertions(&insertions)?;

Tropical Curves

Use tropical geometry for curve counting:

use amari_enumerative::tropical_curves::TropicalCurve;

let curve = TropicalCurve::new(degree, genus);
let count = curve.tropical_count()?;

Modules

Mathematical Background

Chow Rings

The Chow ring A*(X) captures the intersection theory of a variety X:

A*(P^n) = Z[H] / (H^(n+1))

where H is the hyperplane class.

Schubert Cells

The Grassmannian Gr(k,n) has a cell decomposition by Schubert cells:

Gr(k,n) = ⊔ Ω_λ

indexed by partitions λ ⊂ (n-k)^k.

Gromov-Witten Theory

GW invariants count curves via:

⟨τ_a1(γ1),...,τ_an(γn)⟩_{g,β} = ∫_{[M̄_{g,n}(X,β)]^{vir}} ψ_1^{a1} ev_1*(γ1) ∧ ...

Classic Enumerative Problems

Performance

• Parallel Computation: Rayon-based parallelization
• GPU Acceleration: WebGPU for large intersection computations
• Sparse Matrices: Efficient representation of Schubert classes
• Batch Processing: Process multiple curves simultaneously

License

Licensed under either of Apache License, Version 2.0 or MIT License at your option.

Part of Amari

This crate is part of the Amari mathematical computing library.