Moduli of vector bundles on primitive multiple schemes
Jean-Marc Drézet
TL;DR
The paper develops a systematic deformation-theoretic framework for vector bundles on primitive multiple schemes $Y$ with reduced support $X$, introducing a canonical obstruction class $\Delta(\mathbb{E})$ in $\mathrm{Ext}^2_{{\mathcal{O}}_X}(E,E\otimes L^n)$ that governs extension from $X_n$ to $X_{n+1}$ via the fundamental class $\nabla_0(\mathbb{E})$. It constructs fine moduli spaces $M_{n+1}$ for bundles on $X_{n+1}$ whose restriction lies in a given open set $\chi_n$ of $X_n$, showing that $M_{n+1}$ is an affine bundle over the extendable locus $N_n\subset M_n$, with a universal description through a vector bundle and a projective bundle, and connecting this to extensions of line bundles and the Picard groups. The work develops detailed machinery using Čech cohomology, Ext spectral sequences, and universal families, and applies it to Picard groups, products of curves, and moduli on primitive curves (ribbons), including explicit criteria for when moduli spaces fail to be vector bundles due to nontrivial obstructions. It also provides several new primitive multiple schemes with trivial dualizing sheaf and analyzes how these obstructions shape the geometry of moduli across multiplicities. Overall, the paper extends the deformation-theoretic modulation of vector bundles to the nonreduced realm, yielding precise moduli constructions and new phenomena for primitive multiple curves and their Picard groups.
Abstract
A primitive multiple scheme is a Cohen-Macaulay scheme $Y$ such that the associated reduced scheme $X=Y_{red}$ is smooth, irreducible, and that $Y$ can be locally embedded in a smooth variety of dimension $\dim(X)+1$. If $n$ is the multiplicity of $Y$, there is a canonical filtration $X=X_1\subset X_2\subset\cdots\subset X_n=Y$, such that $X_i$ is a primitive multiple scheme of multiplicity $i$. The simplest example is the trivial primitive multiple scheme of multiplicity $n$ associated to a line bundle $L$ on $X$: it is the $n$-th infinitesimal neighborhood of $X$, embedded in the line bundle $L^*$ by the zero section. The main subject of this paper is the construction and properties of fine moduli spaces of vector bundles on primitive multiple schemes. Suppose that $Y=X_n$ is of multiplicity $n$, and can be extended to $X_{n+1}$ of multiplicity $n+1$, and let $M_n$ a fine moduli space of vector bundles on $X_n$. With suitable hypotheses, we construct a fine moduli space $M_{n+1}$ for the vector bundles on $X_{n+1}$ whose restriction to $X_n$ belongs to $M_n$. It is an affine bundle over the subvariety $N_n\subset M_n$ of bundles that can be extended to $X_{n+1}$. In general this affine bundle is not banal. This applies in particular to Picard groups. We give also many new examples of primitive multiple schemes $Y$ such that the dualizing sheaf $ω_Y$ is trivial.