Objective
I propose to study the geometry of Severi varieties. Roughly speaking, Severi varieties parameterize (irreducible) curves of a given geometric genus in a linear system on an algebraic surface. Introduced by Severi in the 1920s, Severi varieties have been intensively studied (mostly in the plane case and in characteristic zero) by many algebraic geometers, such as Zariski, Fulton, Harris, Ran, Kontsevich, Caporaso, Vakil, and Mikhalkin. Despite this intensive study of Severi varieties, many questions have remained open, e.g. the case of positive characteristic, for which neither the dimension nor the irreducibility is known and for which enumerative formulas are also not known. In characteristic zero, the irreducibility property is not known for any surfaces other than the projective plane and Hirzebruch surfaces. (The irreducibility property is, however, known for rational curves in some cases.) In this study, I propose to investigate the geometry of Severi varieties on toric surfaces. In positive characteristic, I intend to: prove dimension formula; construct examples of toric surfaces admitting reducible Severi varieties and classify all such surfaces; describe the geometry of a general curve of a given geometric genus on a toric surface (note that as opposed to the case in characteristic zero, such a curve need not be nodal, and the description of its geometry will include the classification of its singularities); and generalize Mikhalkin's tropical enumerative formulas to this case (it should be emphasized that Severi varieties are not defined over the integers, and their degrees depend on the characteristic). In characteristic zero, I intend to prove the irreducibility property for Severi varieties on toric surfaces. The main tools that will be developed and applied in this research are drawn from deformation theory of schemes, morphisms, and stacks; and from toric and tropical geometries (see part B of the proposal for details).
Fields of science (EuroSciVoc)
CORDIS classifies projects with EuroSciVoc, a multilingual taxonomy of fields of science, through a semi-automatic process based on NLP techniques. See: The European Science Vocabulary.
CORDIS classifies projects with EuroSciVoc, a multilingual taxonomy of fields of science, through a semi-automatic process based on NLP techniques. See: The European Science Vocabulary.
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Programme(s)
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Multi-annual funding programmes that define the EU’s priorities for research and innovation.
Topic(s)
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Calls for proposals are divided into topics. A topic defines a specific subject or area for which applicants can submit proposals. The description of a topic comprises its specific scope and the expected impact of the funded project.
Call for proposal
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Procedure for inviting applicants to submit project proposals, with the aim of receiving EU funding.
FP7-PEOPLE-2009-RG
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Funding Scheme
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Funding scheme (or “Type of Action”) inside a programme with common features. It specifies: the scope of what is funded; the reimbursement rate; specific evaluation criteria to qualify for funding; and the use of simplified forms of costs like lump sums.
Coordinator
84105 Beer Sheva
Israel
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