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Jean-Marc Drézet

Publications and source records attributed to Jean-Marc Drézet.

8 recordsLinked to original sources
arXiv2026

Coherent sheaves on primitive multiple schemes

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 $I_X$ is the ideal sheaf of $X$ in $Y$ and $Y\not=X$, then $L=I_X/I_X^2$ is a line bundle on $X$, called the associated line bundle of $Y$. Even if $X$ is projective, $Y$ needs not to be quasi projective. We define in every case the reduced Hilbert polynomial $P_{red,O_X(1)}(E)$ of a coherent sheaf $E$ on $Y$, depending on the choice of an ample line bundle $O_X(1)$ on $X$. If $E$ is a flat family of sheaves on $Y$ parameterized by a smooth curve $C$, then $P_{red,O_X(1)}(E_c)$ does not depend on $c\in C$. We study flat families of sheaves in two important cases: the families of quasi locally free sheaves, and if $n=2$ those of balanced sheaves. Balanced sheaves are generalizations of vector bundles on $Y$, and could be used to expand already known moduli spaces of vector bundles on $Y$. When $X$ is a smooth projective surface, and $Y$ is of multiplicity 2 we study the simplest examples of balanced sheaves: the sheaves $E$ such that there is an exact sequence \[0\longrightarrow I_P\otimes L\longrightarrow E\longrightarrow I_P=E_{|X} \longrightarrow 0 \ , \] where $I_P\subset O_X$ is the ideal sheaf of a point $P\in X$. They can also be described as the ideal sheaves $E$ of subschemes of $Y$ concentrated on $P$, and such that $E_P$ is generated by two elements whose images in $O_{X,P}$ generate the maximal ideal. There is a moduli space for such sheaves, which is an affine bundle on $X$ with associated vector bundle $T_X\otimes L$ (where $T_X$ is the tangent bundle of $X$). The associated class in $H^1(X,T_X\otimes L)$ can be determined.

math.AG↗
arXiv2026

Moduli of vector bundles on primitive multiple schemes

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.

math.AG↗
arXiv2025

Reducible deformations and smoothing of primitive multiple curves

A primitive multiple curve is a Cohen-Macaulay irreducible projective curve $Y$ that can be locally embedded in a smooth surface, and such that $C=Y_{red}$ is smooth. In this case, $L={\mathcal I}_C/{\mathcal I}_C^2$ is a line bundle on $C$. This paper continues the study of deformations of $Y$ to curves with smooth irreducible components, when the number of components is maximal (it is then the multiplicity $n$ of $Y$). We prove that a primitive double curve can be deformed to reduced curves with smooth components intersecting transversally if and only if $h^0(L^{-1})\not=0$. We give also some properties of reducible deformations in the case of multiplicity $n>2$.

math.AG↗
arXiv2024

Blowing-ups of primitive multiple schemes

A primitive multiple scheme is a Cohen-Macaulay scheme $\bf X$ such that the associated reduced scheme $X={\bf X}_{red}$ is smooth, irreducible, and that $\bf X$ can be locally embedded in a smooth variety of dimension $\dim(X)+1$. If ${\bf I}_X$ is the ideal sheaf of $X$, $L={\bf I}_X/{\bf I}_X^2$ is a line bundle on $X$, called the associated line bundle of $\bf X$. 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. A subscheme $Z$ of ${\bf X}$ is called good if $Z_{red}$ is smooth and connected, and if $d={\rm codim}(Z)$ and ${\bf I}_{Z,{\bf X}}$ is the ideal sheaf of $Z$, then for every closed point $z\in Z$, ${\bf I}_{Z,{\bf X},z}$ can be generated by $d$ elements. Two kinds of subschemes $Z$ of ${\bf X}$ will be considered: the closed smooth subschemes of $X$, seen as subschemes of $\bf X$, and the good subschemes. In the two cases, the blowing-up ${\bf B}_{Z,{\bf X}}$ of ${\bf X}$ along $Z$ is a primitive multiple scheme of multiplicity $n$, and its underlying smooth scheme is the blowing-up ${\bf B}_{Z_{red},X}$ of $X$ along $Z_{red}$. Additional results are obtained in the case of hypersurfaces or points of $X$. We treat the case of $X={\rm P}_2$, with $Z$ a single point $P$. Let $p:\widetilde{\rm P}_2\to{\rm P}_2$ be the blowing-up of ${\rm P}_2$ along $P$. We find all primitive double schemes $\bf X$, $\bf Y$, with ${\bf Y}_{red}=\widetilde{\rm P}_2$, ${\bf X}_{red}={\rm P}_2$, such that there is a morphism ${\rm Y}\to{\rm X}$ inducing $p$ and an isomorphism ${\bf Y}\backslash p^{-1}(P)\to{\bf X} \backslash\{P\}$. We obtain in this way the list of all K3-carpets with underlying smooth variety $\widetilde{\rm P}_2$.

math.AG↗
arXiv2015

Paramétrisation des courbes multiples primitives

The primitive curves are the multiple curves that can be locally embedded in smooth surfaces (we will always suppose that the associated reduced curves are smooth). These curves have been defined and studied by C. Banica and O.Forster in 1984. In 1995, D. Bayer and D. Eisenbud gave a complete description of the double curves. We give here a parametrization of primitive curves of arbitrary multiplicity. Let Z_n=spec(\C[t]/(t^n)). The curves of multiplicity n are obtained by taking an open cover (U_i) of a smooth curve C and by glueing schemes of type U_i X Z_n using automorphisms of U_{ij} X Z_n that leave U_{ij} invariant. This leads to the study of the sheaf of nonabelian groups G_n of automorphisms of C X Z_n that leave the reduced curve invariant, and to the study of its first cohomology set. We prove also that in most cases it is the same to extend a primitive curve C_n of multiplicity n to one of multiplicity n+1, and to extend the quasi locally free sheaf D_n of derivations of C_n to a rank 2 vector bundle on C_n.

math.AG↗
arXiv2011

Faisceaux cohérents sur les courbes multiples

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.

math.AG↗