Algebraic and Arithmetic Geometry Curriculum

82 weeks · ~6.5 hrs/wk · ~530 hrs total Profile: undergraduate algebra background, goal of Berkeley/Harvard PhD qualifying exam level Focus: algebraic constructions and their geometric avatars — every definition paired with a geometric picture

Overview

Phase	Weeks	Theme	File
I	1–20	Classical Varieties (Shafarevich)	Phase I
II	21–42	Scheme Theory (Hartshorne I-II + Mumford)	Phase II
III	43–64	Cohomology and Curves (Hartshorne III-IV)	Phase III
IV	65–82	Arithmetic Geometry (Silverman + Milne)	Phase IV

Commutative algebra is not a separate phase. Study the relevant Atiyah-Macdonald (A&M) sections at the point of first need — each week flags exactly which chapters are prerequisites. Eisenbud’s Commutative Algebra with a View Toward Algebraic Geometry is the secondary CA reference: its geometric commentary on standard theorems is invaluable.

Qualifying exam problem sets used as benchmarks: - Ritvik Ramkumar (Berkeley, 2017): scheme theory + commutative algebra + algebraic topology - Will Fisher (Berkeley, 2024): scheme theory + category theory + algebraic topology - Harvard AG qualifying problem collection: curves, divisors, cohomology, elliptic curves

Dependency Map

flowchart TD
    subgraph P1["Phase I: Classical Varieties (Wks 1-20)"]
        affine["Affine Varieties
Nullstellensatz"]
        projV["Projective Varieties
Homogeneous Coords"]
        maps["Morphisms
Rational Maps"]
        local["Local Geometry
Smoothness, DVRs"]
        curvesCl["Classical Curves
Divisors, Genus, RR"]
    end

    subgraph P2["Phase II: Scheme Theory (Wks 21-42)"]
        spec["Spec A
Structure Sheaf"]
        shvs["Sheaves
Locally Ringed Spaces"]
        schm["Schemes
Morphisms"]
        proj["Proj
Fiber Products"]
        coh["Coherent Sheaves
Picard Group"]
        diff["Differentials
Blowing Up"]
    end

    subgraph P3["Phase III: Cohomology and Curves (Wks 43-64)"]
        derived["Derived Functors
Sheaf Cohomology"]
        cech["Cech Cohomology
H^i of P^n"]
        sd["Serre Duality
Vanishing Theorems"]
        rr["Riemann-Roch
Linear Systems"]
        ellG["Elliptic Curves
j-invariant, Group Law"]
    end

    subgraph P4["Phase IV: Arithmetic Geometry (Wks 65-82)"]
        ellA["EC over Fields
Isogenies, Torsion"]
        mw["Mordell-Weil
over Q"]
        fq["EC over Fq
Frobenius, Hasse"]
        weil["Weil Conjectures
Zeta Functions"]
    end

    affine --> projV
    affine --> maps
    projV --> maps
    maps --> local
    local --> curvesCl

    affine --> spec
    projV --> proj
    local --> shvs
    curvesCl --> coh

    spec --> shvs
    shvs --> schm
    schm --> proj
    proj --> coh
    coh --> diff

    shvs --> derived
    coh --> derived
    coh --> cech
    derived --> cech
    cech --> sd
    sd --> rr
    rr --> ellG
    curvesCl --> rr

    ellG --> ellA
    ellA --> mw
    ellA --> fq
    fq --> weil
    rr --> weil

References

Text	Role
Shafarevich, Basic Algebraic Geometry Vol 1	Phase I primary
Reid, Undergraduate Algebraic Geometry	Phase I companion
Gathmann, Algebraic Geometry lecture notes (free)	Phase I–II problem supplement
Atiyah-Macdonald, Introduction to Commutative Algebra	CA reference (woven throughout)
Eisenbud, Commutative Algebra with a View Toward AG	CA supplement (geometric commentary)
Hartshorne, Algebraic Geometry	Phase II–III primary
Mumford, The Red Book of Varieties and Schemes	Phase II geometric companion
Vakil, The Rising Sea (free)	Phase II reference and exercises
Serre, Faisceaux Algébriques Cohérents (FAC)	Phase III historical source
Miranda, Algebraic Curves and Riemann Surfaces	Phase III curves supplement
Weibel, Introduction to Homological Algebra	Phase III derived functor reference
Silverman, The Arithmetic of Elliptic Curves	Phase IV primary
Ireland-Rosen, A Classical Introduction to Modern Number Theory	Phase IV finite fields
Milne, Lectures on Étale Cohomology (free)	Phase IV Weil conjectures
Fulton, Algebraic Curves (free)	Optional: deeper classical curves reference

Qualifying Exam Sources

Source	Notes
Ritvik Ramkumar, Berkeley 2017	Major: AG + Comm. Algebra; Minor: Alg. Topology
Will Fisher, Berkeley 2024	Major: AG + Category Theory; Minor: Alg. Topology
Harvard AG Qualifying Problems	Broad problem set; use as weekly diagnostic