[Paper Review] On the Göttsche Threshold

Steven L. Kleiman, Vivek Shende+1 more|arXiv (Cornell University)|Apr 27, 2012

Algebraic Geometry and Number Theory27 references5 citations

TL;DR

This paper establishes a sharp upper bound for the Göttsche threshold—the minimal degree of a line bundle on a rational surface for which the degree of the Severi variety of δ-nodal curves is given by a universal polynomial in Chern classes. By relaxing the standard δ-very ampleness condition to a codimension condition on three key loci (nonreduced, non-immersed, or highly singular curves), the authors prove that Göttsche’s conjectured threshold of ⌈δ/2⌉ + 1 holds for ℙ² and Hirzebruch surfaces, and a similar bound applies to classical del Pezzo surfaces.

ABSTRACT

For a line bundle L on a smooth surface S, it is now known that the degree of the Severi variety of cogenus-d curves is given by a universal polynomial in the Chern classes of L and S if L is d-very ample. For S rational, we relax the latter condition substantially: it suffices that three key loci be of codimension more than d. As corollaries, we prove that the condition conjectured by Göttsche suffices if S is P^2 or S is any Hirzebruch surface, and that a similar condition suffices if S is any classical del Pezzo surface.

Motivation & Objective

• To determine the precise conditions under which the degree of the Severi variety of δ-nodal curves on a rational surface is given by a universal polynomial in Chern classes.
• To relax the standard δ-very ampleness condition, which is sufficient but overly restrictive, by identifying weaker, geometrically meaningful conditions.
• To verify Göttsche’s conjectured threshold value of ⌈δ/2⌉ + 1 for ℙ² and Hirzebruch surfaces, and extend the bound to classical del Pezzo surfaces.
• To establish that the Severi variety of δ-nodal curves is open and dense in the full Severi variety under these relaxed conditions, ensuring enumerative invariance.

Proposed method

• Introduce a closed subset V ⊂ |ℒ| consisting of curves that are nonreduced, have components with −K·D₁ ≤ 0, or have nonimmersed components with −K·D₁ = 1.
• Prove that if codim(V) > δ, then the Severi variety |ℒ|δ has codimension δ and is smooth outside V, ensuring the degree is well-defined.
• Extend V to include curves with components of multiplicity ≥3 and −K·D₁ ≤ 3, or multiple components with specific tangency and self-intersection conditions.
• Use deformation theory and the analysis of the tangent map to show that general curves in |ℒ|δ ∖ V are nodal, by ruling out higher singularities via dimension arguments.
• Apply results from deformation theory (e.g., Prp. 15 and Prp. 17) to bound the dimension of families of curves and ensure equigeneric deformations.
• Use the fact that the image of the deformation map γ is constrained by the tangent space to the surface and the normal bundle, leading to a contradiction if singularities exceed δ-nodal type.

Experimental results

Research questions

• RQ1 For which line bundles ℒ on a rational surface S is the degree of the Severi variety of δ-nodal curves given by a universal polynomial in c₁(S), c₁(ℒ), and c₂(S)?
• RQ2 Can the standard δ-very ampleness condition be relaxed while preserving the universality of the degree formula?
• RQ3 Does Göttsche’s conjectured threshold of ⌈δ/2⌉ + 1 for ℙ² and Hirzebruch surfaces hold as an upper bound for the degree formula to be valid?
• RQ4 What geometric conditions on the curve locus ensure that the Severi variety of δ-nodal curves is open and dense in the full Severi variety?

Key findings

• For S = ℙ² and ℒ = 𝒪(d), the degree formula deg|ℒ|δ₊ = Gδ(S,ℒ) holds if d ≥ ⌈δ/2⌉ + 1, confirming Göttsche’s conjecture for ℙ².
• For any Hirzebruch surface, if the loci of nonreduced curves, non-immersed curves, and curves with high-multiplicity or tangency singularities all have codimension > δ, then deg|ℒ|δ₊ = Gδ(S,ℒ).
• For classical del Pezzo surfaces, a similar codimension condition on the same three loci implies the universal degree formula holds.
• The Severi variety |ℒ|δ₊ is open and dense in |ℒ|δ whenever the codimension of the bad locus V exceeds δ, ensuring nodal curves dominate.
• The proof relies on deformation-theoretic arguments showing that general curves in |ℒ|δ ∖ V are nodal, by contradiction via dimension bounds on the image of the deformation map.
• The results are sharp in the sense that the bound is not tight for the first Hirzebruch surface (the blowup of ℙ² at a point), showing the condition is not always necessary.