Bezout's theorem (without proof) and its applications (Cayley-Bacharach theorem). This syllabus is for guidance purposes only :
#Algebraic geometry how to
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. A classic example is the origin of the curve defined by the equations x 3-y 2=0.DRPS : Course Catalogue : School of Mathematics : Mathematics This applies not least to the study of singular points on geometric objects, which are, broadly speaking, the points where the object is not "smooth". The translation to algebra means that algebraic geometry is more suitable for studying geometric problems of higher complexity than other nearby fields. On the other hand, modern algebraic geometry is not so concerned about the equations that describe the objects, but concentrates on the abstract properties of the geometric objects by assigning them algebraic structures. If we look further at all the points the satisfy the two equations x 2+y 2+z 2-1=0 and x+y+z=0, then we get the circle at an angle marked in red on the figure. For example, the two-dimensional sphere can be defined in the three-dimensional Euclidean space R 3 as the set of all points (x,y,z) that satisfy x 2+y 2+z 2-1=0.
The objects we study in algebraic geometry are algebraic varieties, which we can say er geometric objects that can be defined by solution sets of polynomials. Today, algebraic geometry is an area with geometry with connections to other areas such as commutative algebra, complex analysis, topology and number theory in mathematics, cryptography in informatics and string theory in physics. This meant that one could go back and forth between an algebraic treatment of equations for a geometric object and a direct geometric study of the object. A milestone was Descartes' introduction of coordinates in geometry. Algebraic geometry has a long history that can be said to go back to the Euclidean geometry in ancient Greece.
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