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Topics and textbooks
Approximate list of topics: - Nullstellensatz and Zariski topology - Affine and projective varieties - Morphisms and rational functions - Sheaves, ringed spaces, abstract varieties and schemes - Spec and Proj - Smoothness - Dimension - Hilbert theorem on degree and dimension - Bernstein-Kushnirenko theorem, Newton polytopes and a touch of toric varieties - Line bundles, divisors and Picard group (time permitting) - Riemann-Roch theorem on curves (time permitting) A partial list of textbooks: - Milne, Algebraic Geometry (online notes) - Hartshorne, Algebraic Geometry - Shafarevich, Basic Algebraic Geometry (vol. I and II) - Griffith and Harris, Principles of Algebraic Geometry (topological and complex differential geometrical approach) - Karen Smith et al, An Invitation to Algebraic Geometry (elementary introduction) - Cutkosky, Introduction to Algebraic Geometry (new text, online version available at Pitt library for Pitt users) - Harris, Algebraic Geometry, A first course (hands on approach with lots of examples) - Cox, Little, O'Shea, Varieties, ideal and algortihms - Cox, Little, O'Shea, Using algebraic geometry - Cox, Little, Scheck, Toric varieties (each chapter contains a nice quick review of algebraic geometry concepts needed) |
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What's new!
o I will not follow a specific textbook but lot of the topics we cover are from Milne. o We will have a take-home final exam. It will be sometime in the last week of April. o Jan. 9: This week we covered Nullstellensatz and Zariski topology from Milne Chapter 2. o Jan. 22: We have covered Chap. 2 Sections a-l (Noether normalization). We also prove Going up theorem from Chap. 1. o HW1 posted. Due in 2 weeks. o Jan. 29: We started with projective varieties. We covered material from Milne Chap. 6 Sections a, b, c, d. You can also check Hartshorne Chap. 1 Section 2. I will wrap up projective varieties next time and then we will start with sheaf of regular functions, morphisms etc. o Feb. 7: We have been covering projective varieties from Milne Chap. 6. o Feb. 7: HW2 is now posted. o Feb. 17: We started with Chapter 3 of Milne. We covered Sections (a)-(c). We will continue with the rest of Chapter 3 next time. Soon we introduce functors Spec and Proj. o March 14: Most likely no classes this week (March 16-20, Pitt has extended Spring Break to this week). I am still figuring out what teleconferencing app to use to have live lectures (Zoom?). o March 14: HW3 posted! o March 23: We had our first lecture on Zoom today. We covered Hartshorn Chap I, Sec. 5. We will cover some left over material from Chap I, Sec. 4 of Hartshorn and prove that any irreducible variety is birational to a hypersurface. Lecture notes are posted below. o March 25: We finished the proof that any irr. variety is birational to a hypersurface. We then talked about Nakayama's lemma and showed that a subset of maximal ideal m is a generating set iff its image in m/m^2 is a vector space spanning set. This leads to the notion of a system of parameters. We also mentioned completion of a ring and Cohen Structure Theorem in passing. Next time we talk about the case of curves and 1-dim regular local rings. o March 25: I posted my informal notes from beginning of the course until before March Break. o March 27: Friday was a makeup session. We reviewed completion of a ring and Cohen structure theorem (without proof) which states that completion of local ring of a non-singular point is a power series ring. We discussed notion of (discrete) valuation and stated (without proof) the main theorem about regular local rings of dimension 1. They coincide with discrete valuation rings. We gave some examples of curves to illustrate the theorems. o March 30: We talked about "normalization" of a variety. We showed that normalization of a curve is a non-singular variety. o April 1: HW4 posted (not an April's Fool joke!) o April 1: Discussed the theorem that a normal variety is "non-singular in codim 1". Introduced notion of vanishing of a rational function along a prime divisor. Briefly discussed birational classification of algebraic curves. o April 6: We talked about Algebraic Geometry vs Analytic Geometry (so-called Serre GAGA theorems), then we started with definition of blowup of affine space at the origin. Next time we discuss the blowup of a variety along a subvariety and notions of Rees algebra and blowup algebra in commutative algebra. o April 6: There will be an extra lecture this Friday April 10 at 2pm. o April 8: We talked about "tautological line bundle" on projective space, in connection to blow up of affine space at the origin. We reviewed general definition of a line bundle on a variety. We discussed blow up of a variety X along a subvariety Y and related commutative algebra notions of blow-up algebra or Rees algebra, and associated graded algebra (of an ideal). Finally, we started with the notion of "degree" of a projective variety and stated Bezout's theorem. o April 10: We continued with discussing notion of degree of a projective variety. We stated BKK theorem (Bernstein-Khovanskii-Kushnirenko). We defined Hilbert function and polynomial of a projective variety and stated Hilbert's theorem. o April 13: We discussed proof of BKK theorem. o April 15: Finished proof of BKK theorem. Also discussed couple of HW4 problems. Next time, overview of def. of abstract variety and scheme (we have lectures on Monday and Wednesday). Take-home final during Thursday-Friday next week. o April 20: We discussed sketch of proof of Hilbert's theorem. Then started with notion of abstract variety, separation axiom and how to glue affine varieties to get an abstract variety. One more lecture left on Wednesday and we will discuss schemes and Proj construction. o April 22: We finished with def. of abstract variety, scheme and Proj construction. o April 23: Final exam posted (below). If you notice a serious typo let me know. Email me with any questions you have. |
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Lecture notes and homework
Homework 1 Homework 2 Homework 3 March 23 lecture March 25 lecture My informal notes March 27 lecture March 30 lecture Homework 4 April 1 lecture April 6 lecture April 8 lecture April 10 lecture April 13 lecture April 15 lecture April 20 lecture April 22 lecture Final exam |
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Course Information
o Time: MW 2:00PM - 3:15PM o Location: Thackeray 627 o There will homeworks and a final exam. |
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Instructor's
Information
Email: kavehk AT pitt.edu Office: Thackeray #424 Office hours: by appointment Office phone: 412-624-8331 |