[Paper Review] Faisceaux coh\'erents sur les courbes multiples

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

Algebraic Geometry and Number Theory9 citations

TL;DR

This paper introduces generalized rank and degree invariants for coherent sheaves on non-reduced curves embedded in smooth surfaces, enabling a Riemann-Roch theorem on such curves. It defines quasi locally free sheaves, proves every coherent sheaf is quasi locally free on a nonempty open subset, and fully classifies torsion-free sheaves on double curves as reflexive sheaves of the form $I_{2,Z} \otimes L$, with applications to moduli spaces having multiple components.

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 develop foundational invariants—generalized rank and degree—for coherent sheaves on non-reduced curves that are locally embedded in smooth surfaces.
• To establish a Riemann-Roch theorem using these invariants, extending classical results to non-reduced settings.
• To define and study quasi locally free sheaves as the natural generalization of locally free sheaves on singular or non-reduced curves.
• To classify torsion-free sheaves on double curves ($n=2$), showing they are reflexive and of the form $I_{2,Z} \otimes L$.
• To analyze moduli spaces of stable sheaves of generalized rank 3 on double curves, revealing multiple components, one of which is a multiple structure over the moduli space of vector bundles on the reduced curve.

Proposed method

• Define a canonical filtration $C = C_1 \subset \cdots \subset C_n = Y$ for a multiple curve $Y$, where $C_i$ is the $i$-th infinitesimal neighborhood of the reduced curve $C$.
• Construct the graded sheaf $\mathrm{Gr}(E) = \bigoplus_{i=1}^n E_i / E_{i+1}$ for a coherent sheaf $E$ on $Y$, with $E_i = I_i^C E$.
• Define generalized rank $R(E) = \mathrm{rg}(\mathrm{Gr}(E))$ and generalized degree $\mathrm{Deg}(E) = \deg(\mathrm{Gr}(E))$, both additive under short exact sequences.
• Prove the Riemann-Roch formula: $\chi(E) = \mathrm{Deg}(E) + R(E)(1 - g_C)$, where $g_C$ is the genus of the reduced curve $C$.
• Introduce the notion of quasi locally free sheaves as those locally isomorphic to direct sums of $\mathcal{O}_{C_i}$, and prove every coherent sheaf is quasi locally free on a nonempty open subset.
• Use deformation theory and Ext-group analysis to study moduli spaces of stable sheaves on double curves, particularly for generalized rank 3 and degree $d$.

Experimental results

Research questions

• RQ1 How can generalized rank and degree be defined for coherent sheaves on non-reduced curves, and do they satisfy additivity and Riemann-Roch?
• RQ2 What is the structure of torsion-free sheaves on double curves ($n=2$), and are they reflexive?
• RQ3 Can the moduli space of stable sheaves on a double curve have multiple irreducible components, and if so, what is their geometric nature?
• RQ4 Is there a canonical way to describe the moduli space of stable sheaves of generalized rank 3 on a double curve in terms of the moduli of vector bundles on the reduced curve?
• RQ5 What conditions ensure a coherent sheaf on a multiple curve is quasi locally free on a nonempty open subset?

Key findings

• The generalized rank and degree are additive in short exact sequences of coherent sheaves on multiple curves.
• The Riemann-Roch formula $\chi(E) = \mathrm{Deg}(E) + R(E)(1 - g_C)$ holds for all coherent sheaves on a multiple curve $Y$.
• Every coherent sheaf on a multiple curve $Y$ is quasi locally free on some nonempty open subset of $Y$, generalizing the notion of local freeness.
• For double curves ($n=2$), torsion-free sheaves are reflexive and precisely those of the form $I_{2,Z} \otimes L$, where $Z \subset C$ is a finite subscheme 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 of which is a multiple structure over the moduli space of stable vector bundles of rank 3 and degree $d$ on the reduced curve $C$.
• A simple criterion for quasi local freeness is given: a coherent sheaf $E$ on $Y$ is quasi locally free if and only if the restriction $E|_{C_i}$ is locally free for all $i$.