Algebraic Geometry: Deformation Theory with the View of Mori Theory

2026/2027

Academic Year

ENG

Instruction in English

6

ECTS credits

Delivered by:

Faculty of Mathematics

Type:

Optional course (faculty)

When:

1, 2 module

Instructor

Zhgoon, Vladimir

Course Syllabus

Abstract

"The aim of the course is to give the introduction to the advanced topics of algebraic geometry which are usually omitted in the standard course. The fruitfull schematic approach of Grothendieck allows to construct such objects as: Hilbert scheme (which classifies the subschemes of a given scheme), Quot scheme and scheme of morphisms of two schemes. The deformation theory studies the infinitesimal structure of these schemes, that allows to say much about the objects where the deformations are considered. This idea was used by Mori who applied the geometry of rational curves to study the birational geometry of projective varieties. In the course we shall discuss these variety of topics."

Learning Objectives

• Освоение методов теории деформаций для следующих объектов: подсхем в фиксированной схеме, локально свободных пучков на схеме, морфизмов между двумя схемами, абстрактные деформации схемы. Освоение методов теории Мори, связанной с геометрией рациональных кривых, а также основных ее теорем: теоремы о конусе, связности Шокурова, теоремы о свободе линейной системы. Освоение теорем Фултона-Хансена о связности и приложения этих теорем для доказательства теорем Зака.

Expected Learning Outcomes

• Освоение инструментов таких как теоремы связности Фултона Хансена и их применение к геометрии проективных алгебраических многообразий
• Освоение построения схемы Гильберта
• Освоение процедуры вычисления касательных пространств к пространствам морфизмов.
• Освоение техники деформации рациональных кривых
• Освоение техники пучков мультипликаторов для доказательства теорем связности и обращения в нуль

Course Contents

• Deformation theory. Deformations of different objects: schemes, sheaves, morphisms etc. Tangent spaces to the space of deformations. Infinitesimal obstructions.
• Hilbert, Quot, Hom and Chow schemes.
• Applications to the spaces of rational curves. Bend and break technique.
• Multiplier ideals. Kawamata-Viehweg vanishing theorem. Shokurov non-vanishing and base-point-freeness theorem. Mori cone theorem.
• Fulton-Hansen connectedness theorem and its applications to geometry of projective varieties. Zak theorems.

Assessment Elements

• Экзамен
• Проблемные листки

Interim Assessment

• 2026/2027 2nd module
  1/10(2/3(final exam%)+1/2(problem sheets%))

Bibliography

Recommended Core Bibliography

• Deformation theory, Hartshorne, R., 2009
• Positivity in algebraic geometry I : classical setting: line bundles and linear series, Lazarsfeld, R., 2004
• Rational curves on algebraic varieties, Kollar, J., 2009

Recommended Additional Bibliography

• Birational geometry of algebraic varieties, Kollar, J., 2008
• Lectures on resolution of singularities, Kollar, J., 2007
• Local cohomology : a seminar given by A. Grothendieck Harvard University, Hartshorne, R., 1967
• Positivity in algebraic geometry II : positivity for vector bundles, and multiplier ideals, Lazarsfeld, R., 2004

Authors

• Zhgun Vladimir Sergeevich