[Paper Review] Small toric degenerations of Fano threefolds

Sergey Galkin|arXiv (Cornell University)|Sep 7, 2018

Algebraic Geometry and Number Theory22 references3 citations

TL;DR

This paper classifies smooth Fano threefolds that admit small toric degenerations to toric Fano threefolds with ordinary double points, identifying 40 families (4 with Picard number 1, 16 with ρ=2, 16 with ρ=3, and 8 with ρ=4). The classification is based on invariants such as Fano index, degree, Betti numbers, and discriminant, and the method leverages mirror symmetry and I-series computations via Laurent polynomials on singular toric varieties to reconstruct Gromov–Witten invariants of the smoothings.

ABSTRACT

We classify smooth Fano threefolds that admit degenerations to toric Fano threefolds with ordinary double points.

Motivation & Objective

• To answer Batyrev's question on which non-toric smooth Fano threefolds admit small toric degenerations.
• To identify all smooth Fano threefolds that smooth nodal toric Fano threefolds with ordinary double points.
• To provide a complete classification of such degenerations using invariants like Picard number, degree, Fano index, and discriminant.
• To demonstrate that the I-series of the smoothings can be computed directly from the degeneration using Laurent polynomial models.
• To establish that all such smoothings are rational and satisfy specific cohomological constraints.

Proposed method

• Classify all 3-dimensional terminal Gorenstein toric singularities, which are ordinary double points isomorphic to (xy=zt) in 𝔸⁴.
• Use the fact that small degenerations preserve Picard group isomorphism and have Gorenstein terminal singularities.
• Construct Laurent polynomial models on the singular toric threefolds to compute I-series via constant term expansions.
• Apply the quantum Lefschetz formula to the I-series of Grassmannians to verify results for specific cases like B₅.
• Use the relation between the I-series of the smoothing Y and the π-series of the Laurent polynomial fₜ to reconstruct Gromov–Witten invariants without prior knowledge of Y.
• Verify that the computed I-series match known results for Fano threefolds such as B₅, confirming the classification.

Experimental results

Research questions

• RQ1 Which smooth Fano threefolds admit small toric degenerations to nodal toric Fano threefolds with ordinary double points?
• RQ2 What invariants (Picard number, degree, Fano index, Betti number, discriminant) characterize the Fano threefolds that admit such degenerations?
• RQ3 Can the I-series of the smooth Fano threefold be computed directly from the degeneration using Laurent polynomial models?
• RQ4 Are all such smoothings rational, and do they satisfy specific cohomological constraints?
• RQ5 To what extent can the method of small toric degenerations be generalized to complete intersections in toric varieties or other Gorenstein singularities?

Key findings

• The paper identifies exactly 40 families of non-toric smooth Fano threefolds that admit small toric degenerations: 4 with ρ=1, 16 with ρ=2, 16 with ρ=3, and 8 with ρ=4.
• All such smoothings are rational, and their Betti numbers satisfy b ≤ 3, with b=3 only for V₂.₁₂ and b=2 only for B₄ or V₂.₁₉.
• The I-series of the smoothing Y is computed directly from the Laurent polynomial on the degeneration X, matching known results for B₅ via quantum Lefschetz.
• The computed I-series for the degeneration X is π_f(t) = 1 + 6t² + 114t⁴ + 2940t⁶ + 87570t⁸ + …, which matches the I-series of B₅.
• The classification confirms that only Fano threefolds with deg(Y) ≥ 20 and ρ(Y) ≤ 4 can admit such degenerations.
• The method confirms that the anticanonical section of the smoothing is smooth and that the degeneration preserves key cohomological and birational properties.