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The Caporaso-Harris-Ran degeneration principle: proof and applications

Francesco Bastianelli, Ciro Ciliberto, Thomas Dedieu, Concettina Galati, Margherita Lelli-Chiesa, Edoardo Sernesi

TL;DR

This work develops the Caporaso–Harris–Ran degeneration framework for Severi varieties, providing a complete proof of the Caporaso–Harris recursion via a detailed treatment of degenerations, limit linear systems, and deformations of curves on surfaces. It integrates semistable degenerations, the conductor and equisingularity theory, and a robust deformation-theory toolkit to describe limit Severi varieties and their multiplicities, including in the logarithmic setting. The text analyzes explicit degenerations (e.g., quartics to unions of planes) to recover classical counts (Salmon) and to establish when degenerations are well-behaved, culminating in a comprehensive theory applicable to K3 and logarithmic K3 contexts. By connecting deformations of maps with deformations of curves, and by introducing delta-good and well-behaved models, it provides both general dimension bounds and practical criteria for the density of nodal curves in degenerations, with wide implications for curve enumeration on complex surfaces.

Abstract

Severi varieties are the parameter spaces for curves with prescribed homology class and genus on a smooth surface. We describe their limits along degenerations of surfaces, with a view towards the enumeration of curves. This includes a complete proof of the Caporaso-Harris recursive formula, with all the necessary background on deformations of curves and singularities.

The Caporaso-Harris-Ran degeneration principle: proof and applications

TL;DR

This work develops the Caporaso–Harris–Ran degeneration framework for Severi varieties, providing a complete proof of the Caporaso–Harris recursion via a detailed treatment of degenerations, limit linear systems, and deformations of curves on surfaces. It integrates semistable degenerations, the conductor and equisingularity theory, and a robust deformation-theory toolkit to describe limit Severi varieties and their multiplicities, including in the logarithmic setting. The text analyzes explicit degenerations (e.g., quartics to unions of planes) to recover classical counts (Salmon) and to establish when degenerations are well-behaved, culminating in a comprehensive theory applicable to K3 and logarithmic K3 contexts. By connecting deformations of maps with deformations of curves, and by introducing delta-good and well-behaved models, it provides both general dimension bounds and practical criteria for the density of nodal curves in degenerations, with wide implications for curve enumeration on complex surfaces.

Abstract

Severi varieties are the parameter spaces for curves with prescribed homology class and genus on a smooth surface. We describe their limits along degenerations of surfaces, with a view towards the enumeration of curves. This includes a complete proof of the Caporaso-Harris recursive formula, with all the necessary background on deformations of curves and singularities.
Paper Structure (186 sections, 66 theorems, 472 equations, 30 figures)

This paper contains 186 sections, 66 theorems, 472 equations, 30 figures.

Table of Contents

  1. Limits of nodal curves: generalities and examples
  2. The basic question
  3. Classical results
  4. Semistable degenerations
  5. Limit linear systems
  6. Severi varieties and their limits
  7. Example: Limits of 1--nodal plane sections for surfaces in $\mathbf{P}^ 3$
  8. Example: Limits of 3--nodal plane sections for quartic surfaces in $\mathbf{P}^ 3$
  9. Degeneration to two quadrics
  10. Degeneration to a cubic and a plane
  11. Curves in the hyperplane bundle
  12. Curves in the twisted hyperplane bundle: plane quartics
  13. A non-good case
  14. Degeneration of cubics to three planes
  15. Construction of a good model
  16. ...and 171 more sections

Key Result

Lemma (3.6)

Assume $f: S \to \mathbf{D}$ is semistable. Let $Q,Q'$ be smooth irreducible components of $S_0$ intersecting along the double curve$R$. Then where a triple point is the intersection $R\cap Q"$ with a component $Q"$ of $S_0$ different from $Q,Q'$.

Figures (30)

  • Figure 1: Degeneration into two quadrics
  • Figure 2: Degeneration into a cubic plus a plane
  • Figure 3: Limit hyperplane linear system for a degeneration into a cubic and plane
  • Figure 4: the degeneration of the plane
  • Figure 5: the degeneration of a cubic to the union of three planes
  • ...and 25 more figures

Theorems & Definitions (173)

  • Remark (3.3)
  • Example (3.4): Families of surfaces in $\mathbf{P}^3$
  • Example (3.5): Families of polarised K3 surfaces
  • Lemma (3.6): Triple Point Formula
  • Remark (4.7)
  • Example (4.8): See gh-degen
  • Remark (5.9)
  • Definition (5.11)
  • Proposition (5.13): concettina1concettina2, see 04b.concettina
  • Remark (5.14)
  • ...and 163 more