Modification of Simpson moduli spaces of 1-dimensional sheaves by vector bundles [Elektronische Ressource] : an experimental example / Oleksandr Iena

Modiflcation of Simpson moduli spaces of 1-dimensionalsheaves by vector bundles, an experimental exampleOleksandr IenaVom Fachbereich Mathematik der Technischen Universit˜at Kaiserslauternzur Verleihung des akademischen Grades Doktor der Naturwissenschaften(Doctor rerum naturalium, Dr. rer. nat.)genehmigte Dissertation1. Gutachter: Prof. Dr. Gun˜ ther Trautmann2.hter: Prof. Dr. Alexander TikhomirovVollzug der Promotion 07.05.2009D 386iiOleksandr Iena. \Modiflcation of Simpson moduli spaces of 1-dimensional sheaves by vectorbundles, an experimental example".Abstract. Thisthesisdealswiththefollowingquestion. Givenamodulispaceofcoherentsheaves on a projective variety with a flxed Hilbert polynomial, to flnd a natural constructionthatreplacesthesubvarietyofthesheavesthatarenotlocallyfreeontheirsupport(wecallsuchsheaves singular) by some variety consisting of sheaves that are locally free on their support.WeconsiderthisproblemontheexampleofthecoherentsheavesonP withHilbertpolynomial23m+1.Given a singular coherent sheaf F with singular curve C as its support we replace F bylocallyfreesheavesE supportedonareduciblecurveC [C ,whereC isapartialnormalization0 1 0of C and C is an extra curve bearing the degree of E. These bundles resemble the bundles1consideredbyNagarajandSeshadri(cf.[19], [20], [26]). Manypropertiesofthesingular3m+1sheaves are inherited by the new sheaves we introduce in this thesis (we call them R-bundles).