Submitted by ANA KARLA PEREIRA RODRIGUES (anakarla_@hotmail.com) on 2017-08-25T13:37:53Z No. of bitstreams: 1 arquivototal.pdf: 1699241 bytes, checksum: 2f092c650c435ae41ec42c261fd9c3af (MD5) / Made available in DSpace on 2017-08-25T13:37:53Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 1699241 bytes, checksum: 2f092c650c435ae41ec42c261fd9c3af (MD5) Previous issue date: 2017-03-10 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In thisthesis,westudycertaindegenerations/specializationsofthegenericsquare matrix overa eld k of characteristiczeroalongitsmainrelatedstructures,suchthe determinantofthematrix,theidealgeneratedbyitspartialderivatives,thepolarmap de ned bythesederivatives,theHessianmatrixandtheidealofsubmaximalminorsof the matrix.Thedegenerationtypesofthegenericsquarematrixconsideredhereare: (1) degenerationby\cloning"(repeating)avariable;(2)replacingasubsetofentriesby zeros, inastrategiclayout;(3)furtherdegenerationsoftheabovetypesstartingfrom certain specializationsofthegenericsquarematrix,suchasthegenericsymmetric matrix andthegenericsquareHankelmatrix.Thefocusinallthesedegenerations is intheinvariantsdescribedabove,highlightingonthehomaloidalbehaviorofthe determinantofthematrix.Forthis,weemploytoolscomingfromcommutativealgebra, with emphasisonidealtheoryandsyzygytheory. / Nesta tese,estudamoscertasdegenera c~oes/especializa c~oesdamatrizquadradagen erica sobre umcorpo k de caracter sticazero,aolongodesuasprincipaisestruturasrela- cionadas, taiscomoodeterminantedamatriz,oidealgeradoporsuasderivadasparci- ais, omapapolarde nidoporessasderivadas,amatrizHessianaeoidealdosmenores subm aximosdamatriz.Ostiposdedegenera c~aodamatrizquadradagen ericacon- siderados aquis~ao:(1)degenera c~aopor\clonagem"(repeti c~ao)deumavari avel;(2) substitui c~aodeumsubconjuntodeentradasporzeros,emumadisposi c~aoestrat egica; (3) outrasdegenera c~oesdostiposacimapartindodecertasespecializa c~oesdamatriz quadrada gen erica,taiscomoamatrizgen ericasim etricaeamatrizquadradagen erica de Hankel.Ofocoemtodasessasdegenera c~oes enosinvariantesdescritosacima, com destaqueparaocomportamentohomaloidaldodeterminantedamatriz.Paratal, empregamos ferramentasprovenientesda algebracomutativa,com^enfasenateoriade ideais enateoriadesiz gias.

Dans cette thèse, nous interprétons géométriquement la torsion de l'algèbre symétrique d'un faisceau d'idéaux I_Z d'un schéma Z défini par n+1 équations dans une variété n-dimensionnelle. Ceci revient à étudier la géométrie de la projectivisation de I_Z. Les applications de ce point de vue concernent en particulier le domaine des transformations birationnelles de l'espace projectif de dimension 3 au sujet duquel nous construisons des transformations birationnelles explicites qui ont le même degré algébrique que leur inverse, le domaine des courbes libres et presque-libres au sujet duquel nous généralisons une caractérisation des courbes libres en étendant les notions de nombre de Milnor et de nombre de Tjurina. Nous abordons aussi le sujet des hypersurfaces homaloides, notre motivation initiale, au sujet duquel nous exhibons en particulier une courbe homaloide de degré 5 en caractéristique 3. La dernière application concerne le calcul de l'inverse d'une transformation birationnelle. / In this thesis, we interpret geometrically the torsion of the symmetric algebra of the ideal sheaf I_Z of a scheme Z defined by n+1 equations in an n-dimensional variety. This is equivalent to study the geometry of the projectivization of I_Z. The applications of this point of view concern, in particular, the topic of birational maps of the projective space of dimension 3 for which we construct explicit birational maps that have the same algebraic degree as their inverse, free and nearly-free curves for which we generalise a characterization of free curves by extending the notion of Milnor and Tjurina numbers. We tackle also the topic of homaloidal hypersurfaces, our original motivation, for which we produce in particular a homaloidal curve of degree 5 in characteristic 3. The last application concerns the computation of the inverse of a birational map.