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Basic algebraic geometry1/9/2024 ![]() ![]() Shafarevich, Basic Algebraic Geometry (especially Part 1, Chapter 3, Section 3.7) William Fulton, Algebraic Curves: An Introduction to Algebraic Geometry online at I.R. Demonstrate understanding of the proof of the Riemann-Roch theoremįrances Kirwan, Complex Algebraic Curves, LMS student notes.Demonstrate knowledge and understanding of the statement of the Riemann-Roch theorem and an understanding of some of its applications.Demonstrate understanding of the basic concepts, theorems and calculations that relate the zeroes and poles of rational functions with the general theory of discrete valuation rings and divisors on projective curves.Demonstrate understanding of the basic concepts, theorems and calculations related to projective curves defined by homogeneous polynomials of low degree.Footnote to the course notes include (as non-examinable material) references to high-brow ideas such as coherent sheaves and their cohomology and Serre-Grothendieck duality.īy the end of the module the student should be able to: Algebraic varieties have many different types of rings associated with them, including affine coordinate rings, homogeneous coordinate rings, their integral closures, and their localisations such as the DVRs that correspond to points of a non-singular curve. The proof of RR is based on commutative algebra. A middle section of the course emphasizes the meaning and purpose of the theorem (independent of its proof), and give important examples of its applications. The logical relations between these treatments is a little complicated. However, it has many quite different characterisations in analysis and in algebraic geometry, and is calculated in many different ways. In intuitive topological terms, we think of it as the ''number of holes''. The formula involves an invariant called the genus $g(C)$ of the curve. After these preliminaries, most of the rest of the course focuses on the Riemann-Roch space $\mathcal^n$. The first sections establishes the class of non-singular projective algebraic curves in algebraic geometry as an object of study, and for comparison and motivation, the parallel world of compact Riemann surfaces. Leads to: The following modules have this module listed as assumed knowledge or useful background:Ĭontent: The module covers basic questions on algebraic curves. The Riemann-Roch theorem for curves is a first major step towards the classification of algebraic curves, surfaces and higher dimensional varieties, that makes up a large component of modern algebraic geometry, and has applications across the mathematical sciences and theoretical physics. ![]() Synergies: The course is a basic introduction to the study of algebraic varieties (and schemes) and their cohomology. In a similar way, the Cauchy integral theorem is good motivation for the full statement of the Riemann-Roch theorem, although it is not needed for the proof. The idea of meromorphic function from MA3B8 Complex Analysis will be mentioned in explaining the purely algebraic discussion of zeros and poles of a rational function. The idea of integral elements of a number field MA3A6 Algebraic Number Theory is a good warm-up for integral closure that will be used in the course. Useful background: Material such as field extensions and ideals in the polynomial ring from MA3D5 Galois Theory will serve as useful motivation. The definitions and ideas from the first half of MA4A5 Algebraic Geometry are also prerequisites. More specifically, the main technical items are localisation (partial rings of fraction of an integral domain), local rings. As a rough guide, the lectures need the first half of MA3G6 Commutative Algebra. Assessment: 85% exam, 15% assessed worksheetsĪssumed knowledge: Some familiarity with basic ideas of commutative algebra is a prerequisite. ![]()
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