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 |