GSV · The Geometry of Severi varieties on toric surfaces
7РП — „Хора“ (Действия „Мария Кюри“)
- Период
- 2009-10-01 → 2013-09-30
- Финансиране от ЕС
- 100 000 €
- Участници
- 1
- Схема
- MC-IRG
Линиите свързват координатора с партньорите.
Накратко на български
Геометрията на вариетите на Севери върху торни повърхности изследва свойствата на криви с определен род, например колко такива криви преминават през дадени точки. Работата помага за по-доброто разбиране на алгебричната и тропическата геометрия в чистата математика.
Кратко обяснение, генерирано от езиков модел по текста на CORDIS. Оригиналът е по-долу.
Резултати накратко
The Geometry of Severi varieties on toric surfaces
Summary description of the project objectives My research relates to the area of algebraic geometry, a branch of modern pure mathematics, and it includes some aspects of tropical geometry, a combinatorial piece-wise linear geometry. In this project, I study the geometry of Severi varieties on toric surfaces in arbitrary characteristic. I address four main problems, as follows: Problem 1 (Dimension problem) Find the dimension of VΣ,L ,g. Problem 2 (Geometry of curves) Describe the geometry of a general curve C of genus g in |L |, in particular classify the singularities of C. Problem 3 (Enumeration of curves) Find enumerative formulas for the number of curves of genus g in |L | satisfying certain (linear) constraints, e.g., passing through a collection of given points in general position. Problem 4 (Irreducibility problem) Find examples of toric surfaces admitting reducible/irreducible Severi varieties V irr Σ,L ,g in positive characteristic. Classify the toric surfaces Σ for which V irr Σ,L ,g are irreducible. Prove the irreducibility of Severi varieties V irr Σ,L ,g on toric surfaces in characteristic zero. The work performed since the beginning of the project Transfer of knowledge: I gave two graduate courses on Algebraic Geometry, organized students seminar for several terms, an international workshop, and have been organizing a weekly seminar on Algebraic Geometry and Number Theory for the last five years. I also attended several international conferences, and gave many talks on the topic of the project at the conferences and at several seminars in Brazil, France, Germany, Israel, Switzerland, Turkey, and US. Research: I have achieved most of the objectives and technical goals of the project with natural deviations due to the development of my research. The main results are described below.
Текст от CORDIS, на английски · Данни: CORDIS, © Европейски съюз
Цел на проекта
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).
Оригинален текст от CORDIS (на английски).
Участници
- BEN-GURION UNIVERSITY OF THE NEGEV · Beer ShevaКоординаторИзраел
Връзки
Данни: CORDIS, © Европейски съюз