5/3: Homework 7 is due. No classes any more.
5/3: Homework 7 is due. No classes any more.
5/2, Tuesday: Extra office hours: 5-6 p.m. Standard Zoom link for
office hours: Zoom
here.
5/1: Course meeting on Zoom: Chow ring: the examples of the projective
space and projectivization of a vector bundle; Chern
classes. [Notes;
[Hartshorne]: A.2-A.3]
4/28: Course meeting on Zoom: Intersection theory: Chow groups,
continued; An-1(X) = Cl(X) = Pic(X) for n = dim X;
intersection product; the Chow
ring. [Notes;
[Hartshorne]: Appendix A through mid-page 426; Axioms A1, A6, A8 in
Sections A.1-A.2]
4/26: Course meeting on Zoom: Proof of deg D > 2g being a sufficient
condition of very ampleness of 𝒪(D) (separation of tangent
vectors). Intersection theory: Chow groups
Ai(X). [Notes;
[V]: 19.2.9-19.2.11; [Hartshorne]: Appendix A through mid-page 426]
4/24: Homework 6 due. Course meeting on Zoom: A criterion for a closed
embedding. Proof of deg D > 2g being a sufficient condition of very
ampleness of 𝒪(D) (separation of points so
far). [Notes;
[V]: 19.1; [Gathmann 2002/03]: Section 7.7.8]
4/21: Course meeting on Zoom: Another application of Riemann-Roch:
ample and very ample sheaves on
curves. [Slides;
[Gathmann 2002/03]: 7.7.5-7.7.9]
4/19: Course meeting on Zoom: The Riemann-Roch formula and Serre
duality. Applications of
Riemann-Roch. [Slides;
[V = Vakil.04.01.2023]: 18.4.1-18.4.4, 18.5.3-18.5.4]
4/17: No course meeting: I am traveling to the Simons Center for
Geometry and Physics in Stony Brook, NY.
4/14: The topological genus and Euler characteristic. The Euler
characteristic change under a morphism of complex projective
curves. The degree of the canonical divisor of a complex projective
curve as consequence of Riemann-Hurwitz. [[Gathmann 2002/03]: 7.6
through the end]
4/12: The Riemann-Hurwitz formula for curves. [[Gathmann 2002/03]:
7.5.12-7.5.13 and 7.6 through 7.6.2]
4/10: Recap on the ramification index. [Recording of the lecture
available on Canvas. [[V = Vakil.04.01.2023]: 21.7 through 21.7.B;
[Gathmann 2002/03]: 7.5.12-7.5.13]
Sunday, 4/9: Homework 5 due by midnight.
4/7: Divisors on curves. Closed and geometric points. A ramification
point and the ramification index. [[V = Vakil.04.01.2023]: 21.7
through 21.7.B; [Gathmann 2002/03]: 7.5.12-7.5.13]
4/5: Ample and very ample line bundles. [[V = Vakil12.31.2022]: 17.6
through 17.6.3]
4/3: Maps to projective spaces. [[V]: 17.4 through 17.4.3]
3/31: Cartier divisors. [Recording of my lecture available on
Canvas. [Gathmann 2002/03]: 9.3]
3/29: The class group and Picard group. Pic
ℙnk. [[V]: 15.2.8, 15.2.K, 15.2.10,
15.2.13, 15.2.H]
3/27: Homework 4 due. Weil divisors and line bundles, continued:
normal schemes, proof of the injectivity of the map div. [[V]: 15.2.4]
3/24: Weil divisors and line bundles. [[V]: 15.2 through 15.2.7
and Clarification on the homomorphism K(X)
→ K(OX,p), Weil divisors, rational functions and line
bundles]
3/22: DVRs and regularity in codimension 1. [[V]: 13.5 through
13.5.13]
3/20: Weil divisors. [[V]: 15.2.1]
3/17: More on the Hilbert polynomial: it is a
polynomial. Examples. The notion of degree. Bézout's
theorem. [[V]: 19.6.2-19.6.5]
3/15: Recap on projective A-schemes. Sheaves finitely generated by
global sections. Serre's Theorem A. Proof of coherence of cohomology
of coherent shaves on projective schemes. Proof of Serre's vanishing
theorem. [[V]: 16.2.1, 16.2.3, 16.2.7-16.2.8, 19.1.3-19.1.4]
3/13: Remarks on projective A-schemes and Proj. Cohomology of coherent
sheaves on a projective scheme over a Noetherian ring is finitely
generated. Serre's vanishing theorem. The Hilbert polynomial. The
Hilbert function. [[V]: 19.1.3, 19.6 through 19.6.2 (no proofs yet)]
3/6-10: Spring Break. Have a good rest over the break!
3/3: Homework 3 on Chapters 15-16 due by midnight. Proj. Projective
schemes. [[Gathmann 2002/03]: 5.5]
3/1: The genus of a curve. Isomorphism invariants. A glimpse of Serre
duality. [[G]: The rest of Chapter 16]
2/27: Computing the double skyscraper sequence. The Euler
characteristic. [[G]: 16.10-16.12]
2/24: Cohomology computation for twisting sheaves on projective
space. [[G]: 16.9]
2/22: Further properties of cohomology. The long exact
sequence. [Slides;
[G]: 16.7-16.8]
2/20: Motivation of the special case of a separated and compact scheme
X over a field k and a quasi-coherent sheaf ℱ on it for defining
Čech cohomology. First property: H0 (X, ℱ) =
ℱ(X). [[G]: 16.7(a)]
2/17: The conormal sequence. The tangent sheaf. The Čech
complex and sheaf cohomology. [[G]: 15.7, 15.10-15.11, 16.1-16.6]
2/15: Homework on Chapters 14-15 due by midnight. The proof of the
third characterization of the sheaf of differentials. Smoothness of a
variety X is equivalent to local freeness of
ΩX. [[V]: 22.2.20, 22.2.24; [G]: 15.6]
2/13: Recap of the universal property and Euler sequence. The sheaf of
differentials via J/J2. [[V]: 22.2.17-22.2.23; [G]:
15.4-15.5, 15.8]
2/10: Correction of the construction of the sheaf of (relative)
differentials. Its quasi-coherence. The universal property of
differentials. The Euler sequence. [[V]: 22.2.17-22.2.19; [G]:
15.8-15.9]
2/8: Tensoring with locally free sheaves. Kähler
differentials. The cotangent and tangent sheaves on a scheme. [[G]:
Sections 14.20 and 15.1-15.3. [V=Vakil]: Sections 22.1-22.2.8]
2/6: Class meeting and office hours on Zoom only this
Monday. Locally free sheaves (vector bundles), invertible sheaves
(line bundles). The Picard
group. [Slides;
[G]: Chapter 14 through 14.19]
2/3: The adjunction between f* and f*. The case
of a closed embedding f=i. The pullback of a pushforward and the
projection formula. [[Hartshorne]: The pushforward (direct image) and
pullback (inverse image) in the beginning of Section II.5, Exercises
II.1.18 and II.5.1 (d). [G]: Section 14.15]
2/1: Homework on Chapters 13-14 due by midnight. The pullback sheaf
and the fiber. [[G]: Sections 14.13-14.14]
1/30: The pushforward of a quasi-coherent sheaf and the ideal
sequence. [Class notes. [G]: Sections 14.8-14.12]
1/27: The sheaf associated to an R-module, quasi-coherent
sheaves. [Class notes. [G]: Chapter 14 (through 14.7)]
1/25: The relation between a presheaf and its sheafification. Exact
sequences of sheaves, the skyscraper sequence, tensoring twisting
sheaves on a projective space. [Class notes. [G]: The rest of Chapter
13]
1/23: Sheafification. [Class notes. [G]: Chapter 13 through 13.19 (b),
skipping 13.16-13.18 and 13.19 (a)]
1/20: Sheaves of modules, standard constructions: pushforward (direct
image), kernel sheaf, image presheaf, tensor product presheaf. [Class
notes. [G]: Chapter 13 (through 13.13)]
1/18: Introduction. Sheaves of modules. Twisting sheaves on
ℙn. [Syllabus.
Class notes. [G:=Gathmann]: Chapter 13 (through 13.6)]
Last modified: (2023-05-02 17:08:42 CDT)