Algebraic Surfaces
A.Y. 2024/2025
Learning objectives
The aims of this course is to provide the notion of birational morphisms and of minimal model of surfaces, in order to obtain the classification of the algebraic surfaces and to study their geometric properties
Expected learning outcomes
The student will learn the basic results in birational geometry, in particular about the problem of the birational classification of the varieties. Moreover, one will acquire techniques for the construction and the study of projective varieties.
Lesson period: Second semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course can be attended as a single course.
Course syllabus and organization
Single session
Responsible
Lesson period
Second semester
Course syllabus
Background material
Varieties, subvarieties; divisors and line bundles; line bundles, sections, and rational maps.
The canonical bundle; Serre duality.
Positivity
Ample line bundles and their properties: Serre's vanishing and its consequences.
Basics on intersection theory: the definition and first properties; the Néron-Severi group and numerical equivalence; the Hodge index theorem.
Cones and characterisation of positivity properties: Nakai-Moishezon-Kleiman's criterion; growth of cohomology groups; R- and Q-divisors; Kleiman's theorem; the cone of effective curves; the ample and the nef cones; duality between cones and Kleiman's criterion; structure of the nef cone/cone of effective curves; the cone theorem; nefness and ampleness in families.
Big divisors: Iitaka's theorem; big divisors and their characterazions; the cone of effective divisors.
Surfaces
Basics of the theory of projective algebraic surfaces: curves on a surface; the Riemann-Roch theorem, Noether's formula; genus formula.
Birational maps and minimal models: rational maps and linear systems; birational maps; blowing-ups and their properties; birational invariants; Castelnuovo's theorem; minimal models; kodaira dimension and classification though birational invariants.
Examples: surfaces with negative Kodaira dimension: blow ups of P^2 and del Pezzo surfaces.; surfaces with trivial Kodaira dimension: complete intersections, double cover of P^2, elliptic surfaces and Abelian surfaces; surfaces with Kodaira dimension equal to 1: elliptic surfaces; surfaces of general type: complete intersections, double covers of P^2, the Godeaux surface.
Varieties, subvarieties; divisors and line bundles; line bundles, sections, and rational maps.
The canonical bundle; Serre duality.
Positivity
Ample line bundles and their properties: Serre's vanishing and its consequences.
Basics on intersection theory: the definition and first properties; the Néron-Severi group and numerical equivalence; the Hodge index theorem.
Cones and characterisation of positivity properties: Nakai-Moishezon-Kleiman's criterion; growth of cohomology groups; R- and Q-divisors; Kleiman's theorem; the cone of effective curves; the ample and the nef cones; duality between cones and Kleiman's criterion; structure of the nef cone/cone of effective curves; the cone theorem; nefness and ampleness in families.
Big divisors: Iitaka's theorem; big divisors and their characterazions; the cone of effective divisors.
Surfaces
Basics of the theory of projective algebraic surfaces: curves on a surface; the Riemann-Roch theorem, Noether's formula; genus formula.
Birational maps and minimal models: rational maps and linear systems; birational maps; blowing-ups and their properties; birational invariants; Castelnuovo's theorem; minimal models; kodaira dimension and classification though birational invariants.
Examples: surfaces with negative Kodaira dimension: blow ups of P^2 and del Pezzo surfaces.; surfaces with trivial Kodaira dimension: complete intersections, double cover of P^2, elliptic surfaces and Abelian surfaces; surfaces with Kodaira dimension equal to 1: elliptic surfaces; surfaces of general type: complete intersections, double covers of P^2, the Godeaux surface.
Prerequisites for admission
We will assume the necessary basic knowledge in geometry and topology at the level of the courses of a Bachelor's Degree in Mathematics.
In addition, it is strongly recommended that students already have knowledge of complex varieties (alternatively, of algebraic projective varieties over the field of complement numbers), line bundles on them, and their cohomology -- as illustrated, for example, in the course on Complex Varieties or on the Geometry of Schemes of the Master's Degree in Mathematics.
In addition, it is strongly recommended that students already have knowledge of complex varieties (alternatively, of algebraic projective varieties over the field of complement numbers), line bundles on them, and their cohomology -- as illustrated, for example, in the course on Complex Varieties or on the Geometry of Schemes of the Master's Degree in Mathematics.
Teaching methods
In-class lectures, and exercise classes.
Teaching Resources
MyAriel page of the course and materials therein.
The course content is covered in various texts. Here is a subset of them where the topics covered in class can be found and that contain interesting further topics of study.
Part One (Positivity for algebraic varieties):
- R. Lazarsfeld. Positivity in algebraic geometry, Volume 1.
Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge, 48, Springer-Verlag, Berlin, 2004.
- J. Kollár and S. Mori. Birational geometry of algebraic varieties.
Cambridge Tracts in Math., 134, Cambridge University Press, Cambridge, 1998.
Part Two (Classification of algebraic surfaces):
- W. Barth, K. Hulek, C. Peters, A. van de Ven. Compact complex surfaces. Second edition.
Ergebnisse der Mathematik und ihrer Grenzgebiete. Folge, 4. Springer-Verlag, Berlin, 2004.
- A. Beauville, Complex Algebraic Surfaces. Second Edition.
London Math. Soc. Stud. Texts, 34. Cambridge Univ. Press, 1996.
- M. Reid, Chapters on Algebraic Surfaces.
Contained in J. Kollár (ed.), Complex Algebraic Geometry, IAS/Park City Math. Ser., vol. 3, Amer. Math. Soc., Providence R.I., 1997.
The course content is covered in various texts. Here is a subset of them where the topics covered in class can be found and that contain interesting further topics of study.
Part One (Positivity for algebraic varieties):
- R. Lazarsfeld. Positivity in algebraic geometry, Volume 1.
Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge, 48, Springer-Verlag, Berlin, 2004.
- J. Kollár and S. Mori. Birational geometry of algebraic varieties.
Cambridge Tracts in Math., 134, Cambridge University Press, Cambridge, 1998.
Part Two (Classification of algebraic surfaces):
- W. Barth, K. Hulek, C. Peters, A. van de Ven. Compact complex surfaces. Second edition.
Ergebnisse der Mathematik und ihrer Grenzgebiete. Folge, 4. Springer-Verlag, Berlin, 2004.
- A. Beauville, Complex Algebraic Surfaces. Second Edition.
London Math. Soc. Stud. Texts, 34. Cambridge Univ. Press, 1996.
- M. Reid, Chapters on Algebraic Surfaces.
Contained in J. Kollár (ed.), Complex Algebraic Geometry, IAS/Park City Math. Ser., vol. 3, Amer. Math. Soc., Providence R.I., 1997.
Assessment methods and Criteria
The examination will consist of a written part and an oral test.
The written part will consist of a series of exercises that the student must hand in to the lecturer before the oral test takes place, in accordance with the procedures that the lecturer will indicate before each exam session. The exercises will be chosen by the student among those proposed by the lecturer within the exercise sessions of the course. The lecturer will clearly indicate which exercises will be examinable.
The grade of the written part of the exam will make up 30% of the final grade.
During the oral examination, the student will be expected to present some of the results of the course syllabus, and to answer specific questions posed by the lecturer on topics or examples illustrated within the course syllabus.
The grade of the oral examination will make up 30% of the final grade.
The final grade will be expressed on a scale from 0 to 30 with integral increments; 18 is the minimum passing grade. It will be announced immediately upon completion of the oral examination.
The written part will consist of a series of exercises that the student must hand in to the lecturer before the oral test takes place, in accordance with the procedures that the lecturer will indicate before each exam session. The exercises will be chosen by the student among those proposed by the lecturer within the exercise sessions of the course. The lecturer will clearly indicate which exercises will be examinable.
The grade of the written part of the exam will make up 30% of the final grade.
During the oral examination, the student will be expected to present some of the results of the course syllabus, and to answer specific questions posed by the lecturer on topics or examples illustrated within the course syllabus.
The grade of the oral examination will make up 30% of the final grade.
The final grade will be expressed on a scale from 0 to 30 with integral increments; 18 is the minimum passing grade. It will be announced immediately upon completion of the oral examination.
MAT/03 - GEOMETRY - University credits: 6
Lessons: 42 hours
Professor:
Svaldi Roberto
Shifts:
Turno
Professor:
Svaldi RobertoProfessor(s)
Reception:
By appointment (to be agreed upon via email)
Room 2102, Dipartimento di Matematica "F. Enriques", Via Saldini 50