Math 7141 - Autumn 2023: Algebraic Geometry I

[Syllabus]            [References]             [Homework]        [Schedule]         [Lecture Notes]

Syllabus

• Syllabus

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References

The literature on embedded algebraic Varieties is vast. We will be using several references to cover the material discussed in class. Links to electronic copies available through the OSU library are provided whenever possible (access requires connection via an OSU proxy, e.g., by being on campus)

Textbooks and resources:

• Basic Algebraic Geometry (volumes 1 and 2), by Igor Shafarevich (Springer-Verlag, Second edition, 1994).
• Algebraic geometry: a first course by Joe Harris (Springer-Verlag, Graduate Texts in Mathematics Series, GTM 133, 1992).
• Ideals, varieties, and algorithms: an introduction to computational algeraic geometry and commutative algebra by David Cox, John Little and Donal O´Shea (Springer, Undergraduate texts in mathematics Series, fourth edition, 2015). Available online through the OSU library.
• Commutative Algebra: with a View Toward Algebraic Geometry by David Eisenbud (Springer, Graduate Texts in Mathematics Series, GTM 150, 2004). Available online through the OSU library.
• Algebraic Curves and Riemann Surfaces by Rick Miranda (AMS Graduate Studies in Mathematics, Volume 5).
• Introduction to Algebraic Curves by Phillip A. Griffiths (AMS, Translations of mathematics Monographs, Volume 76).

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Homework

There will be regular homework assignements, roughly one every two weeks, posted on the course's website. Participants are encourage to work in teams, but individual solutions must be submitted for grading and credit.

Each student is expected to work on and submit one or two problems per assignment (chosen by the students). Only one solution per problem will be uploaded to Carmen for grading. Students should agree among their peers which problems each of them will submit (and inform me by comments on Carmen to the assignment post).

Solutions should be uploaded as a pdf file (preferrably produced in LaTeX).

• Homework 1: pdf and LaTeX files. Due date: September 8, 2023.
• Homework 2: pdf and LaTeX files. Due date: September 18, 2023.
• Homework 3: pdf and LaTeX files. Due date: October 2, 2023
• Homework 4: pdf and LaTeX files. Two solutions should be uploaded for grading. Due date: October 16, 2023.
• Homework 5: pdf and LaTeX files. Two solutions should be uploaded for grading. Due date: October 30, 2023.
• Homework 6: pdf and LaTeX files. Two solutions should be uploaded for grading. Due date: November 27, 2023.
• Homework 7: pdf and LaTeX files. Due date: December 6, 2023.

As an optional complementary assignment, students are welcome to give a 30 minute presentation in class on a topic not discussed in class. Topics will be selected in agreement with the instructor.

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Tentative Schedule

The following is the schedule of topics that we plan to cover each week (it is subject to change). For a list of topics cover each class, see the corresponding handwritten notes in the section entitled Lectures.

Week	Topics
1	Introduction and overview; affine algebraic varieties and first examples; Zariski topology; Hilbert basis theorem.
2	Ideals defining varieties; Basic duality for affine varieties; irreducible decomposition of affine varieties; primary and irreducible ideals; primary decompositions
3	Minimal and associate primes of ideals; Uniqueness of minimal primary components; Hilbert nullstellensatz
4	Noether Normalization; Coordinate rings of varieties and morphism between affine varieties
5	Function fields; Rational and regular functions on irreducible varieties; characterization of regular maps for algebraically closed fields; sheaf theory
6	Sheaf Theory, sheafification, exact sequences, Introduction to sheaf cohomology; Projective varieties; homogeneous ideals
7	More on projective varieties; graded algebras; projective Nullstellensatz; Homogeneization of affine varieties.
8	Morphisms on projective varieties: rational, regular and projective maps; Veronese morphisms
9	Projective equivalence vs. isomorphisms of projective varieties; Segre morphisms; Products of affine and projective varieties; Abstract prevarieties.
10	Abstract varieties; compactness in algebraic geometry. Topology of products; Finite maps between affine varieties.
11	Examples and main properties of finite maps for affine varieties; Finite maps for abstract varieties; Geometric version of Noether normalization; Dimension of irreducible projective varieties in Pn by means of finite surjective maps to some Pr induced by a linear projection on Pn.
12	Equivalent notiones of dimensions in algebraic geometry: topological dimension of spaces; Krull dimension of rings; Basic properties of rings; Dimension and Noetherianness; Invariance of dimension of affine varieties under finite morphisms; Incomparabity, Lying-over and Going-up theorems.
13	Krull dimension and integral extensions; the dimension of affine space; finitely generated polynomial rings over fields are catenary; dimension of irreducible varieties via transcendence degree; birational invariance of dimension; Krull's Principal ideal theorem and its geometric consequences; dimension of fibers of regular dominant maps between irreducible varieties
14	Tangent spaces; smooth and singular varieties
15	Regular local rings from smooth points on varieties; Smooth points are open and dense; Jacobian criterion for smoothness; irreducible affine varieties are birational to hypersurfaces; first definitions of blow-ups.
16	Main properties of blow-ups; Examples.

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Lectures

• Lecture 1 (Overview of Algebraic Geometry and course description), August 23, 2023.
• Lecture 2 (Affine varieties: basic definitions and first examples; the Zariski topology and its properties, product topology vs. Zariski topology of affine n-dimensional space), August 25, 2023.
• Lecture 3 (Ideals from affine varieties; Basic duality between affine varieties and radical ideals of polynomial rings), August 28, 2023.
• Lecture 4 (Irreducible decompositions of varieties; characterization of irreducible varieties via prime definiting ideals; primary ideals of Noetherian commutative rings), August 30, 2023.
• Lecture 5 (Irreducible ideals of Noetherian commutative rings; irreducible decomposition of proper ideals; existence of minimal primary decompositions on such rings; finiteness of minimal primes of an ideal), September 1, 2023.
• Lecture 6 (Independence of associate primes from minimal primary decomposition; characterization of associated primes via annihilators; uniqueness of minimal primary components), September 6, 2023.
• Lecture 7 (Hilbert's Nullstellensatz I: equivalence between weak and strong versions, proof of weak version using slicing with general coordinate affine hyperplanes), September 8, 2023.
• Lecture 8 (Hilbert's Nullstellensatz II: correspondence between points in Kⁿ and maximal ideals), September 11, 2023.
• Lecture 9 (Noether normalization), September 13, 2023.
• Lecture 10 (Coordinate rings of affine varieties; relative versions of varieties, ideals and relative Nullstellensatz; definition of polynomial morphisms between two affine varieties; correspondence between these maps and K-algebra homomorphism between coordinate rings), September 15, 2023.
• Lecture 11 (Function fields of irreducible varieties, rational maps; definition of regular maps), September 18, 2023.
• Lecture 12 (More on regular maps; algebraic structure on regular maps on irreducible varieties and on regular maps at points), September 20, 2023.
• Lecture 13 (Characterization of the ring of regular functions (at points) over algebraically closed fields; sheaf theory), September 22, 2023.
• Lecture 14 (Sheaf theory II: stalks, topological realizations of sheaves; sections of a sheaf; morphisms of presheaves, isomorphisms of sheaves; checking isomorphisms at the stalk level), September 25, 2023.
• Lecture 15 (Sheaf theory III: sheafification of a presheaf; kernel, image and cokernel presheaves; exact sequences of sheaves; introduction to sheaf cohomology via injective resolutions of sheaves), September 27, 2023.
• Lecture 16 (Projective Varieties I: definition of projective space and its Zariski topology; standard affine decomposition of Pⁿ and standard affine patches; projective subvarieties of Pⁿ and their defining properties), September 29, 2023.
• Lecture 17 (Projective Varieties II: ideals from projective subvarieties of Pⁿ; basic duality for projective varieties; decomposition of projective varieties into irreducibles; the affine cone of a projective variety; the projective Nullstellensatz), October 2, 2023.
• Lecture 18 (Projective Varieties III: proof of the projective Nullstellensatz; the homogeneous coordinate ring; homogeneization and affinization), October 4, 2023.
• Lecture 19 (Projective Varieties IV: The Zariski topology; computing projective closures), October 6, 2023.
• Lecture 20 (Projective Morphisms: function theory on projectie varieties; definition of morphisms, rational and regular maps to A¹ for irreducible projective varieties exploiting these notions on the standard open affine cover of Pⁿ; the sheaf of regular functions on an irreducible projective variety; example: stereographic projection), October 9, 2023.
• Lecture 21 (Projective Morphisms II: constructing regular and rational maps between (irreducible subvarieties of) projective spaces; projective equivalence vs. isomorphisms of projective varieties; examples of regular maps; the Veronese map; example for d=2 and n=1), October 11, 2023.
• Lecture 22 (Projective Morphisms III: Birational varieties, characterization of birational varieties via function fields; the sheaf of global sections on an irreducible variety and its stalks when the field is algebraically closed; the Veronese map), October 16, 2023.
• Lecture 23 (Product of varieties and the Segre embedding: product of affine varieties is an affine variety; Segre embeddings to endow P^m×Pⁿ with the structure of a projective variety; product of projective varieties is a projective variety; the Zariski topology on P^m×Pⁿ; characterization of rational and regular maps P^m×Pⁿ→ P^N), October 18, 2023.
• Lecture 24 (Abstract Varieties I: definition of prevarieties by finite open covers by affine varieties; new prevarieties from old by gluing; main examples: P¹ and the affine line with a double point.), October 20, 2023.
• Lecture 25 (Abstract Varieties II: Product of prevarieties via universal property; restriction to the affine case; Definition of varieties; the affine line with a double point is not a variety; affine varieties are varieties; open and closed subsets of varieties are varieties; the graph of a morphism of varieties is closed), October 23, 2023.
• Lecture 26 (The compactness and Hausdorff condition in Algebraic Geometry: Separateness and completeness; review on the topology of the product of prevarieties; Projective varieties over algebraically closed fields are complete), October 25, 2023.
• Lecture 27 (Completeness and Finite Maps on varieties I: main consequences of completeness of varieties, images of maps of varieties with complete source are closed and complete; no non-constant maps to the projective line from a connected complete variety over an algebraically closed field; infinite irreducible projective varieties over algebraically closed fields cannot avoid hypersurfaces; basic definitions of integral, finite and finite-type morphisms of rings, and how are these properties related), October 27, 2023
• Lecture 28 (Finite maps on varieties II: characterization of sheaves of regular maps on affine varieties over algebraically closed fields via coordinate rings; reduction to dominant maps; example of finite maps; finite morphisms between affine varieties have finite fibers), October 30, 2023.
• Lecture 29 (Finite maps on varieties III: finite maps between affine varieties over algebraically closed fields are closed; geometric interpretation of Noether normalization; definition of finite maps for abstract varieties over algebraically closed fields), November 1, 2023.
• Lecture 30 (Finite maps on varieties IV and linear projections: Equivalent definitions of finite maps between abstract varieties over algebraically closed fields and their main properties; definition of linear projections; Dimension of an irreducible projective variety in Pⁿ by means of a finite surjective map to some P^r induced by a linear projection on Pⁿ.), November 3, 2023.
• Lecture 31 (Dimension Theory I: Basic definitions of topological dimension and Krull dimension of rings; Examples; first properties of dimension and codimension), November 6, 2023.
• Lecture 32 (Dimension Theory II: Dimension of affine varieties is invariant under finite maps; basic properties of Krull dimension; examples of non-Noetherian rings with finite dimension and Noetherian rings with infinite Krull dimension; Incomparability, Lying over and Going-up theorems), November 8, 2023.
• Lecture 33 (Dimension Theory III: Krull dimension is preserved under finite (integral) extensions; K[x₁, …, x_n] is catenary of dimension n for every field K; Dimension of irreducible affine varieties over algebraically closed fields via transcendence degree of function fields; Krull's Principal Ideal Theorem for Noetherian rings; Symbolic powers of prime ideals), November 13, 2023.
• Lecture 34 (Dimension Theory IV: Extension of Krull's Principal Ideal Theorem for non-zero divisors on Noetherian rings; Minimal primes and irreducible components of varieties; Geometric interpretation of Krull's dimension for quasi-affine varieties; generalization of Krull's result to sequences of finitely many regular functions on a quasi-affine variety), November 15, 2023.
• Lecture 35 (Dimension Theory V: A partial converse to the Geometric version of Krull's PIT; controlling the dimension of fibers for regular dominant maps between irreducible quasi-affine varieties; generic behavior of the dimensions of fibers of such maps), November 17, 2023.
• Lecture 36 (Tangent spaces: vector spaces from Noetherian local rings; definition of tangent spaces from stalks of sheaves of global sections; embedded tangent spaces of affine varieties and their defining linear equations; examples; maps on tangent spaces induced by morphisms of varieties; functoriality properties; characterization of such maps using Jacobian matrices), November 20, 2023.
• Lecture 37 (Projective Tangent spaces and smoothness: definition of tangent spaces for projective varieties; definition and functoriality of the differential of a regular map between affine varieties; main theorem: the dimension of the tangent space of an abstract variety at a point is bounded below by the local dimension of the variety at the given point), November 27, 2023.
• Lecture 38 (Regular local rings and smooth points on varieties: definition of Noetherian regular local rings; Noetherian regular local rings are domains; definition of associated graded ring of a Noetherian rign with respect to an ideal; corollary: smooth points to varieties lie in exactly one irreducible component; main theorem: the smooth locus of a variety over an algebraically closed field is open), November 29, 2023.
• Lecture 39 (Smooth points on varieties II, and blow-ups of affine varieties: the smooth locus of a variety over an algebraically closed field is open and dense; irreducible affine varieties are birational to a hypersurface; basic construction of blow-ups of affine varieties), December 1, 2023.
• Lecture 40 (Blow-ups of affine varieties II: The exceptional locus, blow-up of subvarieties, strict and proper transform of subvarieties under the blow-up map, irreduciblity of blow-ups of irreducible varieties, invariance of dimension under blow-ups, equations for blow-ups; the blow-up of affine n-dimensional space at the origin), December 4, 2023.
• Lecture 41 (Blow-ups of affine varieties III: dimension of the exceptional locus; the blow-up only depends on the ideal generated by the chosen functions; extending rational morphism from X to P^N to a regular map on the blow-up; construction of blow-ups of algebraic varieties via gluing blow-ups of an open affine cover; blow-up of projective varities; the tangent cone of a variety at a point; dimension of the tangent cone; examples of curves on the affine plane, their strict transform under the blow-up at the origin, and their tangent cones at the origin), December 6, 2023.

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