Reid's Undergraduate Algebraic Geometry, Chapter I of Hartshorne's Algebraic Geometry and Volume I of Shafarevich's Basic Algebraic Geometry all cover material of this kind.
Classical algebraic geometry, in the sense of the study of quasi-projective (irreducible) varieties over an algebraically closed field, can be studied without too much background in commutative algebra (especially if you are willing to ignore dimension theory).
That said, it is not necessary to learn all of Eisenbud's Commutative Algebra before starting algebraic geometry. (The structure sheaf $\mathscr_X$ in the topos.) If you are willing to restrict yourself to smooth complex varieties then it is possible to use mainly complex-analytic methods, but otherwise there has to be some input from commutative algebra. Indeed, in a very precise sense, a scheme can be thought of as a generalised local ring. The trouble with algebraic geometry is that it is, in its modern form, essentially just generalised commutative algebra. Our group picture from the last day is available to students in the class on request.Reid's Undergraduate Algebraic Geometry requires very very little commutative algebra if I remember correctly, what it assumes is so basic that it is more or less what Eisenbud assumes in his Commutative Algebra! March 16: The Grassmannian and the Hilbert scheme. March 14: Geometry of lines in P3 and the Grassmannian. March 11: Embeddings of curves via Riemann-Roch. March 9: Applications of Riemann-Roch Riemann-Hurwitz. March 4: The divisor associated to a differential on a curve. Algebraic curves and their basic structures.įebruary 21: The local structure of an algebraic curve.įebruary 23: Morphisms between curves and the structure theorem for morphisms of curves.įebruary 25: More on maps between curves quasi-projective varieties and closed maps.įebruary 28: Divisors on curves organizing the function field of a curve. Dimension.įebruary 16: Examples of theorems in the subject birationality to hypersurfaces.įebruary 18: The Nullstellensatz and its proof. Tangent spaces and singular points on hypersurfaces.įebruary 14: Tangent spaces in general the set of singular points is closed. Segre and Veronese maps.įebruary 11: Function field and birationality. įebruary 7: Function theory for projective varieties, rational maps and examples.įebruary 9: Examples of rational maps and birational maps implications for the function field. January 31: Projective space and projective varieties I.įebruary 2: Projective space and projective varieties II.įebruary 4: Eloise lectured on quadrics, the projective Nullstellensatz and the function field. January 28: Morphisms, rational maps, and local rings. January 26: Radical ideals and statement of the Nullstellensatz, morphisms. January 24: Order reversals and vanishing sets, irreducibility, Zariski topology. January 21: Introduction, basic questions, definitions of affine varieties. The material on algebraic curves forms roughly the last third of the course, Fulton's text contains everything necessary, but Kirwan (Complex Algebraic Curves) is very insightful. In the first part of the course, texts of Hulek (Elementary Algebraic Geometry) or Reid (Undergraduate Algebraic Geometry) are excellent, as is Fulton (Algebraic Curves). Texts are always good for different presentations however. There is no textbook that covers exactly the material that we will cover, and the TeX notes form the main resource. Įxample sheets will appear here : 1 2 3 4. There will also be four example sheets which will become available at the usual link. The file will be updated roughly weekly, and will contain slightly more information and may frequently contain references and details that were not provided in lecture. Resources: I will provide TeX lecture notes at this link. We will try to learn the basic language, keep an eye on the richest examples of interest, all with a view towards modern developments.
The material brings together ideas from algebra, topology, and geometry. These objects have been, and remain, the fundamental objects of interest in algebraic geometry. Overview: The course will be an introduction to the basic theory of a ffine and projective algebraic varieties, as well as the theory of algebraic curves.
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