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Maximal Infinitesimal Variation of Hodge Structure for Singular Curves

Mounir Nisse

Abstract

We study the infinitesimal variation of Hodge structure for families of algebraic curves and extend the classical theory from smooth curves to singular and non--planar settings. Using the deformation space $\mathrm{Ext}^1(Ω_X,\mathcal O_X)$ and the dualizing sheaf, we define a singular analogue of maximal infinitesimal variation. For equisingular families of plane curves with planar Gorenstein singularities, we prove that the infinitesimal variation attains maximal rank equal to the arithmetic genus. We show that the rank decomposes into a geometric contribution from the normalization and a singular contribution measured by the $δ$--invariants. For non--equisingular degenerations, the rank defect equals the drop of the total $δ$--invariant and admits an interpretation in terms of vanishing cycles and mixed Hodge structures. We further extend the results to non--planar curves under suitable Petri and deformation conditions.

Maximal Infinitesimal Variation of Hodge Structure for Singular Curves

Abstract

We study the infinitesimal variation of Hodge structure for families of algebraic curves and extend the classical theory from smooth curves to singular and non--planar settings. Using the deformation space and the dualizing sheaf, we define a singular analogue of maximal infinitesimal variation. For equisingular families of plane curves with planar Gorenstein singularities, we prove that the infinitesimal variation attains maximal rank equal to the arithmetic genus. We show that the rank decomposes into a geometric contribution from the normalization and a singular contribution measured by the --invariants. For non--equisingular degenerations, the rank defect equals the drop of the total --invariant and admits an interpretation in terms of vanishing cycles and mixed Hodge structures. We further extend the results to non--planar curves under suitable Petri and deformation conditions.
Paper Structure (21 sections, 168 equations)

This paper contains 21 sections, 168 equations.

Table of Contents

  1. Introduction
  2. Outline of the Paper
  3. Examples and Detailed Computations
  4. Infinitesimal Variation of Hodge Structure for Singular Plane Curves
  5. Comparison: $I$-maximal variation for smooth and singular curves
  6. Application to concrete classes of singularities
  7. IVHS rank for nodal and cuspidal plane curves
  8. Extension to plane curves with higher $\delta$--invariant singularities
  9. Explicit rank defects of the IVHS for concrete non-equisingular degenerations of plane curves
  10. Reinterpretation in the scope of mixed Hodge structures
  11. Non--planar $I$--maximal variation
  12. Non--planar singular $I$--maximal variation
  13. $I$--maximal variation for smooth non--planar curves in three geometric cases
  14. singular non--planar $I$--maximal variation statement of three geometric settings
  15. Singular curves on $K3$ and general type surfaces.
  16. ...and 6 more sections

Theorems & Definitions (34)

  • proof
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  • proof : Proof of Theorem \ref{['thm:nod-cusp']}
  • proof
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  • proof
  • proof
  • ...and 24 more