5/9/18: Last homework is due in Craig Corsi's mailbox.
5/9/18: Last homework is due in Craig Corsi's mailbox.
5/2/18: Last class meeting, but no homework due. Divisors and
the Riemann-Roch Theorem for curves. A glimpse of Serre
duality. [Class notes. Vakil: Sections 14.3, 18.4 through
18.4.3 and 18.5 through 18.5.4. Hartshorne: Sections IV.1,
III.7.6, III.7.12.1-2]
4/30/18: Cohomology of line bundles on projective space. Nonisngular
local rings, schemes, and curves: definitions. [Class
notes. Vakil: Sections 18.1.3, 18.3, and 12.2 through
12.2.B. Hartshorne: Section III.5]
4/25/18: Higher direct images and cohomology. The long exact
sequence. Comparing Grothendieck and Čech
cohomology. Cohomological criterion of affineness. [Class
notes. Hartshorne: Section III.8.1]
4/23/18: Čech cohomology of a quasi-coherent sheaf on an affine
scheme. Cohomology and higher direct images as derived
functors. [Class notes. Vakil: Sections 18.2.4, 23.2, and 23.4
through 23.4.3. Hartshorne: Sections III.1-2, III.8 through III.8.4]
4/18/18: Homework 6 is due. Corrections: Global sections of 𝒪(m)
on ℙ0 and the notion of a constant sheaf vs. a
constant presheaf. More Čech cohomology: the Čech
cohomology of a presheaf on a space. [Class notes. Vakil:
Section 18.2 through 18.2.2.]
4/16/18: Global sections of coherent sheaves on projective
schemes. Introduction to Čech cohomology. [Class
notes. Vakil: Sections 15.4.C-D, 18.2 through
18.2.B. Hartshorne: Theorem II.5.19, Section III.4 through
Lemma 4.1]
4/11/18: Proof of Serre's theorem. (Quasi-)coherent sheaves on Proj R
for a Noetherian graded ring R are associated with graded R-modules
(of finite type, antirespectively). [Class notes. Vakil:
Sections 15.4.F-3. Hartshorne: Section II.5 (pp. 118-119)]
4/9/18: Serre's description of quasi-coherent sheaves on Noetherian
projective schemes. [Class notes. Vakil: Section 15.4 through
15.4.B, 15.4.2-F. Hartshorne: Section II.5 (pp. 118-119),
Corollary II.5.18]
4/4/18: Quasi-coherent sheaves on projective schemes. [Class
notes. Vakil: Sections 15.1-2, 15.3.1. Hartshorne: Section
II.5 (pp. 116-117)]
4/2/18: Homework 5 is due. Blowing up as desingularization. Projective
closure. The projective closure of the elliptic curve y2 =
x3 - x + 1 and the group law on it. [Class
notes. Vakil: Section 19.9.10 (p. 531). Ignore the sheaves
O(D). Hartshorne: Example II.6.10.2 (p. 139). Ignore the
divisors.]
3/28/18: More details on blowing up. [Class notes. Vakil:
Sections 9.3.F, 22.1, 22.3 through 22.3.2, 22.3.4, 22.4.1,
22.4.3. Hartshorne: Sections I.4 (pp. 28-29), II.7
(pp. 163-165)]
3/26/18: Weighted projective spaces. Blowing up. [Class
notes. Vakil: Sections 8.2.11-N, 9.3.F, 22.1, 22.3 through 22.3.2,
22.3.4, 22.4.1, 22.4.3. Hartshorne: Sections I.4 (pp. 28-29),
II.7 (pp. 163-165)]
3/21/18: The Veronese and Segre embeddings. [Class
notes. Vakil: Sections 8.2.6-L and 9.6. Hartshorne:
Exercises I.2.12, I.2.14, I.3.4, I.3.16. II.5.11.]
3/19/18: Homework 4 is due. Example: projection
of Pn\Pn-r-1
onto Pr, r ≤ n. Projective algebraic A-schemes
are algebraic A-schemes. [Class notes. Vakil: Sections 10.1.7-E
(We looked at algebraic A-schemes in class. A k-variety
is just a reduced algebraic k-scheme.) Hartshorne:
Exercise I.3.14]
3/14/18: Spring Break: have a good rest and do not forget to entertain
yourself with the homework.
3/12/18: Spring Break: have a good rest and do not forget to entertain
yourself with the homework.
3/7/18: No class meeting: I am still at the Jim-Murray-fest in
Philadelphia. Use this time to finish the reading assignment and do
some problems from Homework 4, which is due just after the Spring
Break. [Vakil: Sections 10.1.5, 10.1.14, 6.4 through 6.4.D,
8.2.1-B, 8.2.4-G, 8.2.6, 8.2.8-K, 8.2.11-N. Hartshorne: Section
II.2 (pp. 76-77), Exercise II.2.14.]
3/5/18: No class meeting: I am
at Higher Structures 2: A
Conference Honoring Murray Gerstenhaber's 90th & Jim Stasheff's 80th
birthday, University of Pennsylvania, Philadelphia, March 5-8,
2018. Use this time to do reading for the homework, which is due just
after the Spring Break. Reading assignment this time contains some
important things not covered in class. [Vakil: Sections 10.1.5,
10.1.14, 6.4 through 6.4.D, 8.2.1-B, 8.2.4-G, 8.2.6, 8.2.8-K,
8.2.11-N. Hartshorne: Section II.2 (pp. 76-77), Exercise
II.2.14.]
2/28/18: Homework 3 is due. Proj as a scheme: construction of the
structure sheaf. PnA as an
example. [Class notes. Vakil: Sections
4.5.7-11. Hartshorne: Section II.2 (pp. 76-77).]
2/26/18: Closed embedding structures on a closed subset. The
projective spectrum Proj R as a topological space. [Class
notes. Vakil: Sections 4.5.4-7, 8.3.9-G. Hartshorne:
Examples II.3.2.3 and 3.2.6, Section II.2 (pp. 76-77).]
2/21/18: Affine morphisms and spectra of quasi-coherent
algebras. Finite morphisms. Closed embeddings. [Class
notes. Vakil: Sections 7.3.8-L, 8.1. Hartshorne: Sections
II.3 (pp. 84-85), Exercise II.5.17 (b-d).]
2/19/18: Dimension and transcendence degree. Affine
morphisms. [Class notes. Vakil: Sections 11.2,
7.3.3-7. Hartshorne: Section I.1 (pp. 6-7), Exercise II.5.17
(a-b).]
2/14/18: Homework 2 is due (in class or in my mailbox). Class meeting
is canceled and replaced by an Algebraic Geometry Seminar talk
on Canonical Paths on Algebraic Varieties by Professor Daniel
Litt (Columbia University). By the way, an algebraic variety is
just a reduced algebraic scheme over a field, i.e., a separated scheme
of finite type over a field such that sections of the structure sheaf
over open subsets are never nilpotent. Regular classroom, 3:35-4:35
p.m. Abstract: Given a path-connected topological space X and
two points x and y, there is typically no distinguished homotopy
classes of paths between x and y. If X is a normal algebraic variety
over the complex numbers, however, there is a distinguished linear
combination of paths between x and y; there is an analogous statement
for a variety over any local field. I'll make this precise and
describe many applications to arithmetic and geometry: for example, to
explicit descriptions of Galois actions on fundamental groups, and to
the study of the geometry of Selmer varieties. Some of the work
described is joint with Alexander Betts.
2/12/18: Connected components and irreducible ones. Codimension,
height, and localization. [Class notes. Vakil: Sections
3.6.12-13, Q, 11.1. Hartshorne: Sections II.3 (p. 86-87), I.1
(pp. 5-6).]
2/7/18: Closed irreducible sets in Spec R. Irreducible
components. [Class notes. Vakil: Sections 3.6.12, 14-T and 11.1
through 11.1.2. Hartshorne: Sections II.3 (p. 86), I.1
(pp. 5-6).]
2/5/18: The kernel of multiplication. Examples of separated and
unseparated schemes. Separated and quasi-separated
morphisms. Algebraic schemes. Dimension of a scheme. [Class
notes. Vakil: Section 10.1 through 10.1.10. Hartshorne:
Sections II.3 (p. 86) and II.4 (pp. 95-100).]
1/31/18: Separated schemes and morphisms. [Class notes. Vakil:
Section 10.1 through 10.1.C. Hartshorne: Section II.4
(pp. 95-100).]
1/29/18: The first homework is due. Noetherian schemes. Local
vs. global Noetherian conditions. Other finiteness
conditions. [Class notes. Vakil: Sections
3.6.14-S. Hartshorne: Section II.3 (pp. 83-84).]
1/24/18: Fibers: embedding of the fiber in the S-scheme and
examples. Noetherian schemes: main definitions and reminder.
[Class notes. Vakil: Sections 9.3.2-3,
3.6.14-21. Hartshorne: Section II.3 (pp. 87-90).]
1/22/18: More on fibered products: existence, examples. Base change,
fibers of a morphism. [Class notes. Vakil: Sections 9.1-3
through 9.3.1. Hartshorne: Section II.3 from Definition on
p. 87 through p. 88.]
1/17/18: Introduction. Syllabus handed out. K-Points of
projective spaces. Fibered products.
[Syllabus.
Class notes. Vakil: Sections 4.4.9-10. Hartshorne:
Section II.3 (Definition on p. 87).]
Last modified: (2018-05-02 15:02:13 CDT)