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Gorenstein liaison: using commutative algebra to study projective schemes
Responsable du projet Elisa Gorla
   
Collaborateur Maike Massierer
Matey Mateev
Alexandru Constantinescu
   
Résumé The project topics lie within the fields of commutative algebra and algebraic geometry, with a particular focus placed on the study of geometric objects (namely, projective schemes) using tools from (commutative) algebra. My research plans for the upcoming years are mainly driven by relevant theoretical questions in algebraic geometry, and more precisely, by questions related to liaison theory. The theory of liaison or linkage formally started in the seventies, although it had been used by others before, in an hoc manner. Roughly speaking, liaison aims at understanding the class of projective schemes, by partitioning it into families of schemes (the liaison classes) that can all be ultimately ``linked'' to the same scheme. A linkage step consists of taking the union of the scheme that we study with another one, so that the union belongs to a well-studied family of schemes (complete intersections or arithmetically Gorenstein schemes). In an ideal situation, the scheme that we study is linked to one that we understand better, and their union is simpler than each of the two parts. A main focus is placed on the study and comparison of the different types of liaison: liaison via complete intersections, via arithmetically Gorenstein schemes, and biliaison (which essentially corresponds to liaison via a special class of arithmetically Gorenstein schemes: The twisted anticanonical divisors). I plan to further study the linkage pattern of several families of schemes and, in relation to this, investigate which algebraic and geometric operations (e.g., specialization and deformation) transform a scheme into another one in its linkage class. I also plan to use liaison techniques in a different direction: to establish properties of projective schemes. E.g., I plan to study which algebraic properties can be lifted from a general hyperplane section of a scheme to the scheme itself: Most of the results in this area are established for schemes defined over a field of characteristic zero, and I now plan to concentrate on fields of positive characteristic. This line of research aims at reaching a deeper understanding of liaison classes and of the theory of liaison itself. This approach is also very effective in studying projective schemes and their properties. Both aspects of the project are relevant in the context of algebra and geometry. In recent years, I have expanded my interests to include applied aspects in cryptography and coding theory, with immediate direct impact on the real world and its fast developing modern communication. Powerful computational tools from algebra and geometry play a relevant role also in solving such applied problems.
   
Mots-clés liaison theory, algebraic and cohomological invariants of schemes, ideals of minors and pfaffians, G-biliaison, complete intersection, generalized divisors, arithmetically Gorenstein scheme
   
Type de projet Recherche fondamentale
Domaine de recherche commutative algebra, algebraic geometry
Source de financement Swiss National Science Foundation
Etat Terminé
Début de projet 1-9-2009
Fin du projet 31-12-2013
Contact Elisa Gorla