![]() Local ring at a point, tangent spaces, singularities. Irreducibility, irreducible components, rational maps.ĭimension of fibers. As a result, they have at most d zeros on P1.Ĭontinued from last week. Math Stackexchange answer explaining how homogeneous polynomials in X, Y of degree d factor into d homogeneous linear factors. Regular functions and regular maps on quasi-projective varieties. Projective and quasi-projective varieties. Regular maps between affine algebraic sets, isomorphisms.Ĭategory of affine algebraic sets = Category of nilpotent-free, finitely generated algebras.ĭefinition of abstract algebraic varieties. (Shafarevich 1.2.2, Shafarevich A.9, Gathmann 1.2) The ideal associated to a subset of affine space. (Gathmann Chapter 0, Shafarevich Section 1.2.1) Ideals, Hilbert’s basis theorem, Zariski topology. I will also upload my lecture notes and the workshop handouts here.Īffine space, closed (algebraic) subsets of affine space. It is undergoing changes as the class progresses, so the later weeks may not be accurate. ![]() These exam guidelines and problems are now available here.Here is a preliminary outline of the course. Homework is thus optional, but highly encouraged. For part of the exam, you will be asked to solve a very small number of these problems on the blackboard. Two weeks before the exam, we will release a large list of problems, most of which will be taken from earlier homeworks. ![]() We will not post homework solutions, so it is to the student's advantage to attempt the homework and turn in solutions. This may be repeated as many times as necessary. A student whose solution to a homework problem is marked incorrect may consult the professor or instructors as needed, rewrite the solution, and turn it in again at the following exercise class. It will be graded and available for return at the following exercise class. The homework will be posted on Thursday at 5pm and may be turned in the following Friday (8 days later) at the beginning of the exercise session.Sections II.4-II.5 in The Arithmetic of Elliptic Curves, J. Gathmann p.62 up to Remark 7.26, Milne sections 7a-7d Section 9 (birational maps and blowing up) in Gathmann 8-11 (projective varieties) and Lemma 4.1 in Hartshorne, 17-22 in Harris (skip discussion of projective varieties for now) We record below some reading assignments intended to complement the lectures. Macdonald, Addison-Wesley Series in MathematicsĪnd done many of the exercises, but should not be necessary for general comprehension if you are willing to accept certain facts as black boxes. Introduction to Commutative Algebra, M.F.Qing Liu, Algebraic geometry and arithmetic curves, 2006 paperback edition (available to read. BrasseletĪ course in commutative algebra is an official prerequisite, and it will certainly be helpful if you have (for instance) read Andreas Gathmann, Algebraic geometry, course notes linked here. Fulton, Princeton University Press, Introduction to Toric Varieties, J.-P. Toric varieties: Introduction to Toric Varieties, W.Foundations of Algebraic Geometry, Ravi Vakil.Harris, Graduate Texts in Mathematics, Springer Algebraic Geometry, A First Course, J.Skipping around a bit and referring as needed to other Hartshorne, Graduate Texts in Mathematics, Springer ).įor the additional topics, I have in mind the motivating goal of developing the basic theory of curves and surfaces (particularly intersection theory on the latter) to the extent required to present in the final lectures Weil's proof of the Riemann hypothesis for curves over finite fields. ), differential notions (tangent space, smoothness. ), properties of images of morphisms (Chevalley's theorem, universal closedness of projective varieties. ), some key examples (Segre, Veronese, Grassmannian. Algebraic Geometry is a central subject in modern mathematics, with close connections with number theory, combinatorics, representation theory, differential and. ), sheaves, schemes, and the functor of points perspective, some key geometric notions and constructions (birational maps, blowups, projections, divisors. The core topics include affine and projective varieties, morphisms between them, basic properties of such (irreducibility, dimension, finiteness, degree. We plan to cover some of the core topics concerning algebraic varieties over an algebraically closed field and then some additional ones, depending upon the pace to which the course settles. First lecture: Tuesday, February 17, 2015įirst exercise class: Friday, FebruContent
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