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). Leads to: The following modules have this module listed as assumed knowledge or useful background:Īlgebraic geometry studies solution sets of polynomial equations by geometric methods. Synergies: The following modules go well together with Algebraic 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. 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.
Moreover, Hilbert's Nullstellensatz and primary decomposition will be essential for the foundations.
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Miles Reid, Undergraduate commutative algebra, CUP.Assessment: Assignments (15%), 3 hour written exam (85%)Īssumed 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. Miles Reid, Undergraduate algebraic geometry, CUP.įrances Kirwan, Complex algebraic curves, CUP. The main book for this course will be the book by Miles Reid, Quadric surfaces, blow ups, rational and birational maps. Hilbert Basis Theorem and the Nullstellensatz. Affine varieties and their rings of functions. Bezout's theorem (without proof) and its applications (Cayley-Bacharach theorem). This syllabus is for guidance purposes only :
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have learned how to formulate and prove basic statements about algebraic varieties in precise abstract algebraic languageĪssessment Information See 'Breakdown of Assessment Methods' and 'Additional Notes', above.Īdditional Information Academic description have increased their knowledge of finitely generated commutative rings and their fields of fractions, be familiar with explicit examples including plane curves, quadrics, cubic surfaces, Segre and Veronese embeddings, have knowledge of the basic affine and projective geometries, Students who successfully complete this module will : Summary of Intended Learning Outcomes A first course in algebraic geometry is a basic requirement for study in geometry, algebraic number theory or algebra at the MSc or PhD level. Programme Level Learning and Teaching Hours 2,ĭirected Learning and Independent Learning Hoursīreakdown of Assessment Methods (Further Info) Information for Visiting Students Pre-requisitesĭisplayed in Visiting Students Prospectus?ĭelivery period: 2013/14 Semester 2, Available to all students (SV1)īreakdown of Learning and Teaching activities (Further Info) Students MUST have passed: ( Algebra (MATH10021) AND Numbers & Rings (MATH10023) ) conics, plane curves, quadric surfaces.Įntry Requirements (not applicable to Visiting Students) Pre-requisites morphisms and rational maps between varieties, Hilbert Basis Theorem and the Nullstellensatz, We plan to cover Sections 1-5 and 7 from Reid's book (see Reading List below), which include : The focus will be on explicit concrete examples.
In algebraic geometry: affine and projective varieties, and the mapsīetween them. This course will introduce the basic objects Motivation for further study through the introduction of minimalīackground material supplemented by a vast collection of examples. The goal of the course is to give a basic flavour of the subject as Spaces defined by polynomial equations in several variables.īesides providing crucial techniques and examples to many otherĪreas of geometry and topology, recent decades have seen remarkableĪpplications to representation theory, physics and to the construction of algebraic codes. It is a classical subject with a modern face that studies geometric Undergraduate Course: Algebraic Geometry (MATH11120) Course Outline SchoolĪlgebraic geometry studies geometric objects defined algebraically. DRPS : Course Catalogue : School of Mathematics : Mathematics
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