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Course syllabus - Algebraic Geometry, 5 credits

Information about the course

  • Course code: FOUK034
  • Third-cycle subject: Mathematics/Applied Mathematics
  • School: School for Education, Culture and Communication
  • Responsible department: Department for Mathematics and Physics
  • Valid from:
  • Established by: Dean of School
  • Decision date: 2025-03-04
  • Level of education: Third cycle level

Course objective

The purpose of the course is to give the student the opportunity to learn the foundations of algebraic geometry, in particular affine and projective algebraic geometry, and to give the student the opportunity to study some part of modern algebraic geometry.

Course content

  • Affine algebraic geometry: Affine space, affine algebraic sets, the Zariski topology, the correspondence between algebraic sets and ideals, Hilbert’s Nullstellensatz, irreducible components, affine varieties, the dimension of affine varieties, morphisms of affine varieties, local rings.
  • Affine plane curves: Plane algebraic sets, singularities, tangent lines, discrete valuation rings, local rings, intersection numbers.
  • Projective algebraic geometry: Projective space, graded rings, projective algebraic sets, the Zariski topology on projective space, the correspondence between homogeneous ideals and algebraic sets, irreducible components, projective varieties, the dimension of a projective variety, morphisms of projective varieties, local rings, the relationship between affine and projective varieties.
  • Projective plane curves: Linear systems, Bezout’s theorem, inflection points and the Hessian, polar curves, dual curves, the Cayley-Bacharach theorem.
  • One of the following (each student should choose one): (1) Basic theory of schemes and sheaves (2) Curve theory, e.g. the Riemann-Roch theorem, the Riemann-Hurwitz theorem, elliptic curves (3) Basic theory of surfaces, e.g. cubic surfaces, Del Pezzo surfaces, Riemann-Roch for surfaces, Enriques-Kodaira classification (4) Basic enumerative geometry, e.g. Kontsevich’s formula counting rational curves in the projective plane (5) Basic arithmetic geometry, e.g. rational points on conics, Fermat’s last theorem in small degrees (6) Basic geometry in positive characteristic, e.g. curves over finite fields, (the statement of) the Weil conjectures (7) Own suggestion after discussion with examiner.

Intented learning outcomes

After passing the course the doctoral student should be able to

  1. account for, explain and exemplify the central objects in algebraic geometry such as affine and projective varieties.
  2. state central theorems in algebraic geometry.
  3. use methods from algebraic geometry to solve problems regarding curves and surfaces.
  4. independently study and account for a part of modern algebraic geometry.

The intended qualitative targets in relation to the Higher Education Ordinance, appendix 2.

Knowledge and understanding

For the Degree of Doctor, the doctoral student shall demonstrate:

  • A1: broad knowledge and systematic understanding of the research field as well as advanced and up-to-date specialised knowledge in a limited area of this field, and
  • A2: familiarity with research methodology in general and the methods of the specific field of research in particular.

Competence and skills

For the Degree of Doctor, the doctoral student shall demonstrate

  • B1: the capacity for scholarly analysis and synthesis as well as to review and assess new and complex phenomena, issues, and situations autonomously and critically,
  • B4: the ability in both national and international contexts to present and discuss research and research findings authoritatively in speech and writing and in dialogue with the academic community and society in general, and
  • B6: the capacity to contribute to social development and support the learning of others both through research and education and in some other qualified professional capacity.

Judgement and approach

For a Degree of Doctor the doctoral student shall demonstrate

  • C2: specialised insight into the possibilities and limitations of research, its role in society and the responsibility of the individual for how it is used.

Teaching formats

Lectures and seminars.

Examination

INL1, written assignment, 1 cr, concerning learning outcomes 1-3, grade fail (U) or pass (G).

INL2, written assignment, 1 cr, concerning learning outcomes 1-3, grade fail (U) or pass (G).

INL3, written assignment, 1 cr, concerning learning outcomes 1-3, grade fail (U) or pass (G).

SEM1, seminar, 2 cr, concerning learning outcome 4, grade fail (U) or pass (G).

Grade

Examinations included in the course are assessed according to a two-grade scale, fail or pass.

A person who has not passed the regular examination shall be given the opportunity to retake the test.

Requirements

To participate in the course and the examinations included in the course, the applicant must be admitted to doctoral studies.

Selection criteria

Selection of applicants will be made in accordance with the ranking below.

  1. Doctoral students at Mälardalen University.
  2. Doctoral students at other universities.