12/14: Homework due by the beginning of class. Sheaves of modules.
[[G]: Chapter 13 through 13.13, skipping 13.10 (b), 13.11, and 13.12]
12/14: Homework due by the beginning of class. Sheaves of modules.
[[G]: Chapter 13 through 13.13, skipping 13.10 (b), 13.11, and 13.12]
12/09: Open and closed subschemes. Fiber products. Separated, reduced,
finite-type schemes. Varieties and pre-varieties as schemes. [[G]:
The rest of Chapter 12]
12/07: Locally ringed spaces and their morphisms, schemes. [[G]:
Chapter 12 through 12.31, skipping 12.29]
12/02: Affine schemes, stalks and values of regular
functions. Examples. [[G]: Chapter 12 through 12.21.]
11/30: Schemes, finally, on popular demand: the
definition/construction of Spec R.
[Lecture
slides; [G]: Chapter 12 through 12.17]
11/23: Homework due by the beginning of class. The tangent and
cotangent spaces and smooth varieties. [[G]: Chapter 10 through the
end.]
11/18: Blowing up as extension of morphisms. The tangent space. [[G]:
Section 9.26 and Chapter 10 through 10.1]
11/16: Blowing up as resolution of singularities. [[G]: Chapter 9
(Section 9.17(b) through 9.21, skipping tangent cones)]
11/11: Blowing up: the proper (strict) transform; equations defining
the blowup; blowing up a point in 𝔸n. [[G]: Chapter 9
(Section 9.11 through 9.17(a))]
11/9: Intro to birational geometry. Rational and birational maps. The
function field. Blowing up. [[G]: Chapter 9 through Section 9.11]
11/4: Grassmannians as projective varieties. [[G]: Chapter 8 from 8.14
through the end]
11/2: Homework due at the beginning of class. Exterior
algebra. Grassmannians and the Plücker embedding. [[G]: Chapter 8
through 8.13]
10/28: Projective varieties: closed maps; complete varieties; the
Veronese embedding. [[G]: Chapter 7 from Section 7.16 through the end]
10/26: Projective varieties: the Segre embedding; projective varieties
are varieties; closed maps; complete varieties; the Veronese
embedding. [[G]: Sections 7.9 through 7.15]
10/21: Projective varieties: morphisms; examples (PGL(n+1) = Aut
ℙn, projections from a point; conics in
ℙ2). [[G]: Chapter 7 through Section 7.8]
10/19: Homework due at the beginning of class. Projective varieties:
properties of ℙn as a topological space;
homogenization and projective closure; regular functions; projective
varieties are prevarieties. [[G]: Chapter 6: 6.25 through
the end; Chapter 7 through 7.3]
10/14: Projective varieties: affine cones, the homogenous coordinate
ring, projective Nullstellensatz, projective subvarieties, the Zariski
topology. [[G]: Chapter 6 from Section 6.12 through 6.24]
10/12: Projective space: intro. Graded algebra. Homogeneous ideals and
projective varieties. [[G]: Chapter 6 through Section 6.12]
10/7: Prevarieties: gluing, examples, open and closed subprevarieties,
products. Separation and varieties. [[G]: Chapter 5 from Section 5.4 to the end]
10/5: Homework due at the beginning of class. New viepoint on affine
varieties. A distinguished open set is an affine
variety. Prevarieties: intro. [[G]: Chapter 4 through the end;
Chaptaer 5 through Section 5.3]
9/30: Morphisms. [[G]: Chapter 4 through 4.14]
9/28: The stalk 𝒪X,a is a localization. Ringed
spaces. [[G]: Chapter 3 through the end; Chapter 4 through Section
4.1]
9/23: Homework due date revised to this class meeting. The affine
plane with deleted origin is not a distinguished open set. Sheaves,
including those of regular functions. The stalk and germs. Correction:
no evaluation map ℱa → k, only
𝒪X,a → k. [[G]: Chapter 3 from 3.11 through 3.17]
9/21: Homework due at the beginning of class. Regular
functions. Distinguished open sets. Regular functions over them. [[G]:
Chapter 3 through 3.10]
9/16: Independent reading: the Zariski topology. Do related homework
problems as you read! [[G]: Chapter 2 (through the end)]
9/14: Independent reading: subvarieties of an affine variety; the
Zariski topology. Do related homework problems as you read! [[G]:
Chapters 1 (through the end) and 2 (through 2.24)]
9/9: Proof of the easier parts of Nullstellenstaz. Bijection between
affine varieties and radical ideals. Examples. Bijection between
points and maximal ideals. Properties of the ideals of affine
varieties. The coordinate ring. The idea of Zariski topology. [Class
notes. [G]: Chapters 1 (through 1.15) and 2 (first page)]
9/7: Introduction. Affine algebraic varieties, a.k.a. affine algebraic
sets: the loci V(S) of zeroes of a collection S of polynomials. Their
properties. The ideal I(X) of a subset X of an affine space. First
properties. Hilbert's Basis theorem and Nullstellensatz worded.
[Syllabus.
Class notes. [Gathmann]: Chapters 0 and 1 (through 1.10, not yet discussing the proof)]
Last modified: (2022-12-14 15:57:12 CST)