[Paper Review] Faisceaux cohérents sur les courbes multiples

Jean–Marc Drézet|arXiv (Cornell University)|Feb 24, 2004

Algebraic Geometry and Number Theory29 references19 citations

TL;DR

This paper develops a foundational framework for coherent sheaves on non-reduced curves embedded in smooth surfaces, introducing generalized rank and degree invariants to prove a Riemann-Roch theorem. It characterizes torsion-free sheaves on double curves as reflexive and shows moduli spaces of stable sheaves of generalized rank 3 have multiple components, one of which is a multiple structure over the moduli space of vector bundles on the reduced curve.

ABSTRACT

Multiple primitive curves have been defined and studied by A. Banica and O. Forster. We introduce new invariants for coherent sheaves on these curves and define quasi locally free sheaves, which play the same role as locally free sheaves on smooth varieties. A more detailed study of torsion free sheaves is given for primitive curves of multiplicity 2.

Motivation & Objective

• To establish a coherent sheaf theory on non-reduced curves that are locally embedded in smooth surfaces.
• To define new invariants—generalized rank and degree—for coherent sheaves on such curves.
• To prove a Riemann-Roch theorem using these invariants.
• To characterize torsion-free and quasi locally free sheaves on double curves.
• To analyze the structure of moduli spaces of stable sheaves of generalized rank 3 on double curves.

Proposed method

• Introduce a canonical filtration of coherent sheaves on a multiple curve Y, indexed by the layers C₁ ⊂ C₂ ⊂ ⋯ ⊂ Cₙ = Y.
• Define generalized rank and degree via the associated graded sheaves along this filtration.
• Use canonical filtrations to define quasi locally free sheaves, which are locally isomorphic to direct sums of O_{C_i} sheaves.
• Prove that every coherent sheaf on Y is quasi locally free on a nonempty open subset.
• Analyze ideal sheaves I_{n,Z} of finite subschemes Z on the reduced curve C, especially in the double curve case (n=2).
• Apply deformation theory to study deformations of I_{n,Z} as sheaves on the ambient surface S.

Experimental results

Research questions

• RQ1 How can generalized rank and degree be defined for coherent sheaves on non-reduced curves to extend classical Riemann-Roch?
• RQ2 What is the structure of torsion-free sheaves on double curves, and are they reflexive?
• RQ3 How do moduli spaces of stable sheaves on double curves decompose into components?
• RQ4 What is the role of quasi locally free sheaves in the birational geometry of moduli spaces on multiple curves?
• RQ5 Can the moduli space of stable sheaves on a double curve be related to the moduli space of vector bundles on the reduced curve?

Key findings

• Torsion-free sheaves on double curves (n=2) are reflexive, a key structural property.
• Torsion-free sheaves of generalized rank 2 on C₂ are of the form I_{2,Z} ⊗ L, where Z is a finite subscheme of C and L is a line bundle on Y.
• The moduli space of stable sheaves of generalized rank 3 and degree d on a double curve has multiple irreducible components.
• One component is a multiple structure over the moduli space of stable vector bundles of rank 3 and degree d on the reduced curve C.
• Every coherent sheaf on Y is quasi locally free on a nonempty open subset, generalizing the notion of locally free sheaves on smooth curves.
• A Riemann-Roch formula is established using the generalized rank and degree invariants, extending classical results to non-reduced settings.