Math 8254: Class Outlines

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o 05/09/08: Last HW is due! Algebraic curves and algebraic function fields. Analogy with algebraic number fields. Hurwitz's theorem. [H: Section IV.2]

o 05/07/08: Algebraic curves and algebraic function fields: Riemann's theorem that the function field determines the curve.

o 05/05/08: Application: the genus of a planar curve (continued). Projectivity of complete algebraic curves. [H: Exercise I.7.2b, Section IV.3]

o 05/02/08: Application: the genus of a planar curve (continued). [H: Exercise I.7.2b]

o 04/30/08: Lemma: H0 (X, O) = k. The Riemann-Roch theorem (the case of an arbitrary divisor). Different interpretations. Applications: the degrees of the canonical divisor and a principal divisor; the genus of a planar curve. [H: Sections I.3.4, IV.1, II.6.10, Exercise I.7.2b]

o 04/28/08: Serre duality. Divisors on curves and the Riemann-Roch theorem. [H: Section III.7 (7.1, 7.6, 7.7, 7.12, 7.13), pp. 136-137, 294-295]

o 04/25/08: Rational mappings, complete local systems, Bertini's theorem. [H: pp. 24-26, 157-158, 179]

o 04/23/08: Functorial property of divisors, naturality of the isomorphism Cl (X) = Pic X, total and proper transforms of divisors. [H: pp. 165-166]

o 04/21/08: Remark on quotient singularities. Cartier divisors and invertible sheaves. Complete linear systems. [H: pp. 144-145, 156-157]

o 04/18/08: Cartier divisors and Weil divisors. [H: Proposition 6.11]

o 04/16/08: Divisors: divisors on affine schemes, the cone example, the total fraction ring (sheaf) of a scheme, definition of a Cartier divisor, motivation, principal Cartier divisors. [H: Proposition 6.2, Example 6.5.2, pp. 140-141]

o 04/14/08: Divisors: generic points and function fields, more on principal divisors, the divisor class group, examples. [H: Section II.6 (Lemma 6.1 and through p. 133)]

o 04/11/08: Higher direct images: coherence for a projective morphism. Divisors: Weil divisors, principal divisors. [H: Theorem III.8.8b and Section II.6 (through p. 131)]

o 04/09/08: Higher direct images: quasi-coherence (the case of a quasi-compact and separated morphism, continued; the case of a Noetherian scheme). [H: Section III.8]

o 04/07/08: Higher direct images: definition as the sheaf associated to the cohomology presheaf, quasi-coherence (the case of a quasi-compact and separated morphism). [H: Section III.8]

o 04/04/08: Discussion of homework (the Picard group problem). Criterion for ampleness (continued). [H: Exercise III.4.5, Proposition III.5.3, and Section III.8]

o 04/02/08: Bézout's theorem as an application of sheaf cohomology (continued). Criterion for ampleness. [H: Corollary I.7.8 and Proposition III.5.3]

o 03/31/08: Euler characteristic, examples. Bézout's theorem as an application of sheaf cohomology. [H: Exercises III.5.1, III.5.3, and I.7.2, Corollary I.7.8]

o 03/28/08: Finiteness of cohomology of a projective scheme: Serre's theorem. [H: Theorem III.5.2]

o 03/26/08: Cohomology of Pn, continued: the computation of pairing of H0 with Hn, vanishing of middle cohomology. [H: Theorem III.5.1]

o 03/24/08: Problem Set 3 is due. Grothendieck's vanishing theorem for Pn. Cohomology of Pn: the computation of Hn (Pn, O(-n-1)). [H: Section III.2 and Theorem III.5.1]

o 03/14/08: Long exact sequence for Čech cohomology, Leray's theorem on relation between Čech cohomology and usual sheaf cohomology. The cohomology of P1. [H: Section III.4, Exercise III.4.11]

o 03/12/08: Cohomology of an affine scheme. Čech cohomology. [H: Sections III.3 and III.4, Exercise III.4.4 (a)]

o 03/10/08: Homework discussion. Sheaf cohomology: long exact sequence. [H: Section III.1]

o 03/07/08: Sheaf cohomology: definition, H0, cohomology of flabby sheaves, independence of resolution. [H: Sections III.1 and III.2]

o 03/05/08: Sheaf cohomology: properties of flabby resolutions. [H: Sections III.1 and III.2]

o 03/03/08: Homework 2 is due. Sheaf cohomology: flabby resolutions. [H: Sections III.1 and III.2]

o 02/29/08: Exact sequences of sheaves and flabby (flasque) sheaves. [H: Section II.1.2 and Exercises II.1.2 and II.1.16]

o 02/27/08: The covering space (espace étalé) associated to a sheaf. [H: Exercise II.1.13]

o 02/25/08: Morphisms to Pn. Ample invertible sheaves and projective morphisms. Sheaf[H: Section II.7 (Morphisms of Schemes to Pn and Ample Invertible Sheaves)]

o 02/22/08: On the universal property of P(E): continuation. [H: Section II.7 (Proposition 7.12)]

o 02/20/08: On the universal property of P(E). [H: Section II.7 (Proposition 7.11)]

o 02/18/08: Quasi-coherent sheaves on relative Proj S. [H: The Proj section of II.7]

o 02/15/08: Guest speaker: Professor Duiliu-Emanuel Diaconescu from Rutgers University will speak on "Extremal Transitions and Localization in Gromov-Witten Theory" in our regular classroom, Vincent Hall 313, at the regular time, 2:30 p.m.

o 02/13/08: The projectivization P(E) of a quasi-coherent sheaf on a scheme, relative projective spaces, projective space bundles. Blowup. [H: The Proj section of II.7]

o 02/11/08: The sheaf of ideals of a hypersurface of degree m is O(-m). The relative case: Proj of a sheaf of graded commutative algebras. [H: The first page of the Proj section of II.7]

o 02/08/08: Correspondence between sheaves of modules over Proj S and graded S-modules. Projective schemes and homogeneous ideals. [H: Exercises II.3.12, II.5.9 (c), pp. 119-120]

o 02/06/08: Correspondence between sheaves of modules over Proj S and graded S-modules. [H: pp. 118-119, Exercise II.5.9 (a,b)]

o 02/04/08: Operations on quasi-coherent sheaves on Proj S: tensor products and Hom's. [H: pp. 109, 117, regarding Hom's, see for example, Lemma II.5.7 in M. Miyanishi's "Algebraic Geometry"]

o 02/01/08: I will be out of town at a conference: no class meeting, no office hours.

o 01/30/08: Generalities about gluing sheaves. Quasi-coherent sheaves on Proj S. [H: Exercise II.1.22, pp. 116-117]

o 01/28/08: Morphisms of graded rings and associated morphisms of schemes. Examples of weighted projective spaces and projective schemes. [H: Exercise II.2.14]

o 01/25/08: Proof of the "surprise" proposition. Noetherian graded rings. Weighted projective spaces. [H: Exercises II.5.13 and II.3.12]

o 01/23/08: Introduction and general discussion of the course for this term. Projective schemes: Proj S of a graded ring, the structure of a scheme on it (reviewed). "Surprise" proposition: how a subring or a quotient ring of a graded ring might give rise to the same Proj. Proof to follow. [Hartshorne (H): Pages 76-77, Exercises II.5.13 and II.3.12]


Last modified: Wed Jan 23 17:21:24 CDT 2008