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MA4A5 Algebraic Geometry

Lecturer: Rohini Ramadas

Term(s): Term 1

Status for Mathematics students: List C

Commitment: 30 lectures plus assignments

Assessment: Assignments (15%), 3 hour written exam (85%)

Formal registration prerequisites: None

Assumed knowledge: MA3G6 Commutative Algebra: The Module will make free use of the basic concepts of ring and module theory, ideals, prime and maximal ideals, localisation, integral closure. Moreover, Hilbert's Nullstellensatz and primary decomposition will be essential for the foundations.

Useful background: It may be helpful, though not absolutely essential, to be acquainted with basic notions of projective geometry and in particular the concept of projective space from MA243 Geometry. Furthermore, the notion of the exterior algebra of a vector space introduced for example in MA3H5 Manifolds is useful background, but will be fully recalled.

Synergies: The following modules go well together with Algebraic Geometry:

Leads to: The following modules have this module listed as assumed knowledge or useful background:

Content:
Algebraic geometry studies solution sets of polynomial equations by geometric methods. This type of equations is ubiquitous in mathematics and much more versatile and flexible than one might as first expect (for example, every compact smooth manifold is diffeomorphic to the zero set of a certain number of real polynomials in R^N). On the other hand, polynomials show remarkable rigidity properties in other situations and can be defined over any ring, and this leads to important arithmetic ramifications of algebraic geometry.

Methodically, two contrasting cross-fertilizing aspects have pervaded the subject: one providing formidable abstract machinery and striving for maximum generality, the other experimental and computational, focusing on illuminating examples and forming the concrete geometric backbone of the first aspect, often uncovering fascinating phenomena overlooked from the bird's eye view of the abstract approach.

In the lectures, we will introduce the category of (quasi-projective) varieties, morphisms and rational maps between them, and then proceed to a study of some of the most basic geometric attributes of varieties: dimension, tangent spaces, regular and singular points, degree. Moreover, we will present many concrete examples, e.g., rational normal curves, Grassmannians, flag and Schubert varieties, surfaces in projective three-space and their lines, Veronese and Segre varieties etc.

Books:

  • Atiyah M.& Macdonald I. G., Introduction to commutative algebra, Addison-Wesley, Reading MA (1969)
  • Harris, J., Algebraic Geometry, A First Course, Graduate Texts in Mathematics 133, Springer-Verlag (1992)
  • Mumford, D., Algebraic Geometry I: Complex Projective Varieties, Classics in Mathematics, reprint of the 1st ed. (1976); Springer-Verlag (1995)
  • Reid, M., Undergraduate Algebraic Geometry, London Math. Soc. Student Texts 12, Cambridge University Press (2010)
  • Shafarevich, I.R., Basic Algebraic Geometry 1, second edition, Springer-Verlag (1994)
  • Zariski, O. & Samuel, P., Commutative algebra, Vol. II, Van Nos- trand, New York (1960)

Additional Resources