Courbes multiples primitives et déformations de courbes lisses

2012

Preprint

Resume. A primitive multiple curve is a Cohen-Macaulay irreducible projective curve Y that can be locally embedded in a smooth surface, and such that Y red is smooth.This paper studies the 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 are particularly interested in deformations to n disjoint smooth irreducible components, which are called fragmented deformations. We describe them completely. We give also a characterization of primitive multiple curves having a fragmented deformation.

“…In this case the line bundle on C associated to C n is O C . The following result is proved in [8] (théorème 1.2.1):…”

Section: Simple Primitive Multiple Curvesmentioning

confidence: 98%

Fragmented deformations of primitive multiple curves

2012

Preprint

“…Primitive multiple curves and quasi locally free sheaves (cf. [1], [2], [4], [5], [6], [7], [8], [12]).…”

Section: 1mentioning

confidence: 99%

“…Primitive multiple curves and quasi locally free sheaves (cf. [1], [2], [4], [5], [6], [7], [8], [12]). Let n be the smallest integer such that Y = C (n−1) , C (k−1) being the k-th infinitesimal neighbourhood of C, i.e.…”

Section: 1mentioning

confidence: 99%

Non-reduced moduli spaces of sheaves on multiple curves

2020

Advances in Geometry

Self Cite

“…This case deg(L) = 0 has been completely treated in [10]: a primitive multiple curve of multiplicity n can be deformed in disjoint unions of n smooth curves if and only if I C is isomorphic to the trivial bundle on C n−1 . In [9] it has been proved that this last condition is equivalent to the following: there exists a flat family of smooth curves C → S, parametrised by a smooth curve S, s 0 ∈ S such that C s 0 = C, such that Y is isomorphic to the n-th infinitesimal neighbourhood of C in C. The problem of determining which primitive multiple curves of multiplicity n can be deformed to reduced curves having exactly n components, allowing intersections of the components, is more difficult. A necessary condition is h 0 (L * ) > 0.…”

Section: Deformations To Reduced Reducible Curvesmentioning

confidence: 99%

Reachable sheaves on ribbons and deformations of moduli spaces of sheaves

2017

Int. J. Math.

Self Cite

Resume.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,If Y is a ribbon and h 0 (L −2 ) = 0, then Y can be deformed to smooth curves, but in general a coherent sheaf on Y cannot be deformed in coherent sheaves on the smooth curves.It has been proved in [11] that a ribbon with associated line bundle L such that deg(L) = −d < 0 can be deformed to reduced curves having 2 irreducible components if L can be written as L = O C (−P 1 − · · · − P d ) , where P 1 , · · · , P d are distinct points of C. In this case we prove that quasi locally free sheaves on Y can be deformed to torsion free sheaves on the reducible curves with two components. This has some consequences on the structure and deformations of the moduli spaces of semi-stable sheaves on Y .