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What's new!
o Jan 28: Topics covered so far: - Proof of Hilbert Nullstellensatz (from Milne's algebraic geometry notes) - Review of affine algebraic varieties (from Milne's) - Section 1.1 Affine toric varieties (from Cox-Little-Schenck) - Section 1.2 Convex cones o Feb 22: We are almost finished with 1.3. Next week after proving the criterion for smoothness of an affine toric variety, we will start with projective varieties (review of basic notions about projective varieties). o March 18: We are currently discussing Section 2.2. Today we will talk about normal and very ample polytopes. Before Spring Break we talked about Minkowski sum of convex bodies/polytopes and proved that Minkowski sum is cancelative (using the notion of support function of a convex body). o March 18: HW3 is posted. It is due March 29. o April 11: HW4 posted. It is due April 24. o April 11: We will finish with orbit-cone correspondence next class (Theorem 3.2.6). Next week we will talk about degree of a variety, Hilbert polynomial and BKK theorem (degree of a projective toric variety). We finish the course with a short introduction to tropical geometry and tropical compactifications. o April 11: There will be a take-home exam which will be posted around April 24th with 2 days to complete it. o Final exam will be posted Wednesday evening and will be due by Friday midnight. Please scan your exams or type them up and email them to me. Please make sure they are readable! There might be problems from HWs. I will return the HWs on Wednesday. Topics for the final exam: -Generalities on affine varieties, nullstellensatz, coordinate ring etc. -Algebraic torus, lattice of one-parameter subgroups, lattice of characters -Affine toric varieties, affine toric variety of a finite set, affine toric variety of an affine semigroup, affine toric variety of a r.p.st.c. cone -Toric ideals -Affine toric variety of a face of a cone vs affine toric variety of the cone -We did not discuss rings of invariants (Sec. 1.3, p. 43) and product of varieties (appendix to Chap. 1) but you can use them if you want. -Generalities about polyhedral cones -Generalities on projective varieties, homogeneous ideal, homogeneous coordinate ring. etc. -Lattice points and projective toric varieties -Dimension of projective toric variety associated to a finite set, affine charts of a projective toric variety -Generalities on convex polytopes -Normal and very ample polytopes -Projective toric variety of a polytope -Intersection of affine pieces (p. 82) -Smoothness, normality -Normal toric varieties and fans -Generalities on abstract varieties, sheaves and glueing affine varieties along open subsets -Regular functions and rational functions, structure sheaf of a variety -Fan, smooth, smilpicial, complete -Toric variety associated to a fan -Normal fan of a polytope -Orbit-cone correspondence -Limit of one-parameter subgroups (Prop. 3.2.2) -Sumihiro's theorem and fan associated to a toric variety -Hilbert function and Hilbert polynomial of a projective variety (with an embedding into a projective space), degree of a projective variety, Hilbert's theorem relating degree and leading coefficient of Hilbert polynomial (references: Harris "Algebraic geometry, a first course" and Mumford's "Complex projective varieties") -BKK theorem (Bernstein-Kushnirenko theorem), statement as number of solutions of a generic system of polynomials and as degree of a projective toric variety (one reference Kaveh-Khovanskii "Convex bodies and algebraic equations") -Sketch of proof of BKK theorem o April 24: Final exam posted! |
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Weekly homework and other online material
Take-home final exam (due April 26 midnight) Homework 4 (due April 24) Homework 3 (due Friday March 29) Suggested problems to work on: Section 2.0: Problems 5, 7 Section 2.1: Problems 1, 2, 4, 5, 8 Section 2.2: Problems 1, 3, 6, 13 Homework 2 (due Wednesday March 6) HW1 (due Friday February 15) from Cox-Little-Schenck: Suggested problems to work on: Section 1.0: Problems 1, 6 Section 1.1: Problems 1(a)-(e), 2, 10, 12, 14 Section 1.2: Problems 4, 14, 15 Section 1.3: Problems 3, 4, 11, 14 Problem: Show that T=(C^*)^n is an affine algebraic group. That is, show that the points in T are in 1-1 correspondence with points in an affine algebraic group. Please hand in above problem as well as Sec. 1.0 #6, Sec. 1.1 #2, #14, Sec. 1.2 #14, Sec. 1.3 #11. |
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Course Information
Text: Cox, Little, Schenck Toric Varieties (first 3 chapters) Time of lectures: Monday-Wednesday-Fridays 2:00PM - 2:50PM Location of lectures: 321 Thackeray Hall o There will be homeworks and final exam. This course is an introduction to algebraic geometry through theory of toric varieties, it discusses and related to combinatorics of convex polytopes. o We plan to cover first three chapters of Cox-Little-Schenck. o If time permits we will touch upon tropical geometry, hyperplane arrangements and matroids.
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Instructor's
Information
Email: kaveh AT pitt.edu Office: Thackeray #424 Office hours: By appointment. You can drop by anytime, I am usually in. Office phone: 412-624-8331 |