Phase I — Classical Varieties

Weeks 1–20 · ~130 hrs

Goal: Build the geometric dictionary before scheme theory arrives. Every object defined here has a scheme-theoretic avatar in Phase II — Phase I is about seeing and internalizing the geometry first. By the end, affine and projective varieties, morphisms, smoothness, divisors, and the Riemann-Roch theorem should feel concrete and familiar.

Weeks 1–6 primary: Harvard Math 137 (Algebraic Geometry, Brooke Ullery), 24 lectures + 11 problem sets. Lectures and problem sets: https://people.math.harvard.edu/~bullery/math137/ Primary text for Math 137: Fulton, Algebraic Curves (free at http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf)

Weeks 7–20 primary: Shafarevich, Basic Algebraic Geometry Vol 1, Chapters II–III Intuition companion: Reid, Undergraduate Algebraic Geometry (skim for pictures)

Why this structure: Math 137 replaces Shafarevich §I entirely with a more exercise-dense, pedagogically sequenced treatment using Fulton. It adds two topics Phase I would otherwise lack — intersection numbers and Bézout’s theorem — which pay dividends in Phase III. Shafarevich §II–III (smoothness through elliptic curves) has no equivalent in Math 137 and is covered in Weeks 7–20.

Weeks 1–6 — Harvard Math 137

The 24 lectures and 11 problem sets map naturally to a 6-week block at ~5–8 hrs/week. Work through the lecture notes sequentially; the problem sets are the primary exercise source for this block.

All lecture PDFs and problem sets are at https://people.math.harvard.edu/~bullery/math137/.

Week 1 — Affine Varieties and the Nullstellensatz

Lectures: - Sec 1: What is algebraic geometry? - Sec 2: Algebraic sets - Sec 3: The ideal of a subset of affine space - Sec 4: Irreducibility and the Hilbert Basis Theorem - Sec 5: Hilbert’s Nullstellensatz - Sec 6: Algebra detour - Sec 7: Affine varieties and coordinate rings - Sec 8: Regular maps

Concepts to understand:

What algebraic geometry studies: solution sets of polynomial equations as geometric objects
Algebraic sets $V(f_1, \ldots, f_r) \subset \mathbb{A}^n$; the ideal $I(X)$ of a subset
Hilbert Basis Theorem: every ideal in $k[x_1, \ldots, x_n]$ is finitely generated
Weak and strong Nullstellensatz: $I(V(J)) = \sqrt{J}$ over algebraically closed $k$
Irreducibility; algebraic set is irreducible iff $I(X)$ is prime
Coordinate ring $k[X] = k[x_1, \ldots, x_n]/I(X)$; regular functions; morphisms
Algebra–geometry duality: affine varieties $\leftrightarrow$ finitely generated reduced $k$-algebras

Problem sets: PS1, PS2

Milestone

Compute $\sqrt{(x^2, xy)} \subset k[x,y]$ and verify this equals $I(V(x^2, xy))$. Observe that $V(x^2, xy) = V(x)$ even though $(x^2, xy) \neq (x)$ — the radical accounts for the difference. Explain why the hypothesis that $k$ be algebraically closed cannot be dropped (give a counterexample over $\mathbb{R}$).

Week 2 — Local Rings, Plane Curves, and Intersection Numbers

Lectures: - Sec 9: Rational functions and local rings - Sec 10: Affine plane curves - Sec 11: Discrete valuation rings and multiplicities - Sec 12: Intersection numbers

Concepts to understand:

Rational functions and local rings $\mathcal{O}_{X,P}$: functions defined near $P$
Affine plane curves: tangent lines, multiplicity of a point, branches
Discrete valuation rings (DVRs): uniformizers, valuation $v_P(f)$ = order of vanishing
Intersection number $(C \cdot D)_P$ at a point: via $\dim_k \mathcal{O}_P/(f, g)$
Properties of intersection numbers: symmetry, additivity, invariance under linear equivalence

Problem sets: PS3

Bonus from Math 137: Intersection numbers (Sec 12) are Fulton’s key tool — not covered in Shafarevich at this stage. They foreshadow intersection theory in Phase III and will make Bézout’s theorem feel inevitable.

Milestone

Compute $(C \cdot L)_O$ where $C = V(y - x^2)$ and $L = V(y)$ at $O = (0,0)$: $(C \cdot L)_O = \dim_k \mathcal{O}_O/(y - x^2,\, y) = \dim_k k[x]_{(x)}/(x^2) = 2$. The parabola is tangent to the $x$-axis — the intersection multiplicity of 2 detects the tangency that a naive count of distinct points would miss.

Week 3 — Projective Space and Projective Varieties

Lectures: - Sec 13: Projective space - Sec 14: Projective algebraic sets - Sec 15: Homogeneous coordinate rings and rational functions - Sec 16: Affine and projective varieties

Concepts to understand:

Projective $n$-space $\mathbb{P}^n_k$: homogeneous coordinates $[x_0 : \cdots : x_n]$
Projective algebraic sets: common zeros of homogeneous polynomials
Homogeneous coordinate ring $S(X)$: graded ring, NOT the ring of functions on $X$
Standard affine cover: $\mathbb{P}^n = U_0 \cup \cdots \cup U_n$ with $U_i \cong \mathbb{A}^n$
Affine and projective varieties as a unified notion; quasiprojective varieties

Problem sets: PS4

Milestone

Construct an explicit isomorphism $\mathbb{P}^1 \xrightarrow{\sim} V_+(x_0 x_2 - x_1^2) \subset \mathbb{P}^2$ via $[s:t] \mapsto [s^2 : st : t^2]$, and write down its inverse on the chart $x_0 \neq 0$ as $[x_0 : x_1 : x_2] \mapsto [x_0 : x_1]$. Verify both composites are the identity, and check that the image is exactly the conic $V_+(x_0 x_2 - x_1^2)$.

Week 4 — Morphisms, Projective Curves, and Bézout

Lectures: - Sec 17: Morphisms of projective varieties - Sec 18: Projective plane curves - Sec 19: Linear systems of curves - Sec 20: Bézout’s Theorem

Concepts to understand:

Morphisms of projective varieties: defined by homogeneous polynomials of the same degree
Projective plane curves: degree, tangent lines, singular points, intersection with lines
Linear systems of plane curves of degree $d$: a projective space parameterizing curves
Bézout’s Theorem: two projective plane curves of degrees $d$ and $e$ with no common component meet in exactly $de$ points (counted with intersection multiplicity)

Problem sets: PS5, PS6

Bonus from Math 137: Bézout’s theorem (Sec 20) with proof. This is used repeatedly in the Harvard qualifying exam collection.

Milestone

A smooth cubic $C$ and a smooth conic $Q$ in $\mathbb{P}^2$ with no common component meet in exactly $3 \cdot 2 = 6$ points by Bézout. For $C = V_+(y^2 z - x^3 + xz^2)$ and $Q = V_+(x^2 + y^2 - z^2)$, verify that all intersection points are distinct (multiplicity 1 each) by checking the Jacobians are linearly independent at each solution.

Week 5 — Abstract Varieties, Rational Maps, and Blowing Up

Lectures: - Sec 21: Abstract varieties - Sec 22: Rational maps and dimension - Sec 23: Rational maps of curves - Sec 24: Blowing up a point in the plane

Concepts to understand:

Abstract varieties: gluing affine pieces via transition maps; the correct intrinsic notion
Rational maps $f: X \dashrightarrow Y$: defined on a dense open subset; domain of definition
Birational equivalence: rational inverse in both directions; $k(X)$ is a birational invariant
Dimension via transcendence degree: $\dim X = \text{trdeg}_k k(X)$
Fiber dimension theorem: generic fiber has dimension $\dim X - \dim Y$
Blowing up a point in $\mathbb{A}^2$: $\text{Bl}_0 \mathbb{A}^2 \subset \mathbb{A}^2 \times \mathbb{P}^1$; exceptional divisor $E \cong \mathbb{P}^1$

Problem sets: PS7, PS8

Milestone

Show the Cremona involution $\phi: [x:y:z] \mapsto [yz:xz:xy]$ is birational by verifying $\phi \circ \phi = \text{id}$ wherever both are defined. Identify the three base points $[1:0:0], [0:1:0], [0:0:1]$ where $\phi$ is undefined, and describe the three lines $V(x), V(y), V(z)$ along which the image degenerates.

Week 6 — Math 137 Consolidation

Complete remaining problem sets and consolidate.

Problem sets: PS9, PS10, PS11

Consolidation checklist:

For every affine object (coordinate ring, regular map, rational function, local ring), identify its projective analogue
Work through the Harvard qual collection: all problems tagged “affine variety,” “projective variety,” “morphism,” “rational map”
State Bézout’s theorem and use it to compute intersection numbers for 3 explicit pairs of plane curves
State and prove the Nullstellensatz from scratch without notes

Milestone

You should now be fluent in the language of classical algebraic geometry — affine and projective varieties, morphisms, rational maps, dimension, and intersection numbers. The bridge to Shafarevich §II (which opens with tangent spaces) requires only the language of local rings, which Math 137 covered in Lec 9.

Weeks 7–20 — Shafarevich Vol 1, Chapters II–III

From here the primary text is Shafarevich, Basic Algebraic Geometry Vol 1. Chapters I of Shafarevich is now fully replaced by Math 137 — begin directly at Chapter II.

Note on overlap: Math 137 covered DVRs (Lec 11) and blowing up (Lec 24). Weeks 12–13 below revisit these in greater depth; treat them as consolidation rather than new material.

Week 7 — Tangent Spaces and Smoothness

CA prerequisite: A&M Ch 11 (discrete valuation rings) — read §11.1 this week.

Concepts to understand:

Tangent space $T_{X,P}$ at a point: $(\mathfrak{m}_P/\mathfrak{m}_P^2)^\vee$ — dual of the cotangent space
Jacobian criterion: if $X = V(f_1, \ldots, f_m) \subset \mathbb{A}^n$, then $T_{X,P} = \ker(\partial f_i/\partial x_j)(P)$
A point $P \in X$ is smooth (nonsingular) iff $\dim T_{X,P} = \dim X$
Smooth $\Rightarrow$ local ring is regular; regular local ring has $\dim_k \mathfrak{m}/\mathfrak{m}^2 = \dim \mathcal{O}_{X,P}$
The smooth locus is open and dense in any irreducible variety

Reading:

Shafarevich Vol 1, §II.1–II.2 (~3.5 hrs)

Problems:

Shafarevich II.1 #1, 2, 3, 4, 5
Find the singular locus of $V(y^2 - x^2(x+1)) \subset \mathbb{A}^2$ (nodal cubic)
Find the singular locus of $V(y^2 - x^3) \subset \mathbb{A}^2$ (cuspidal cubic)

Milestone

For the nodal cubic $X = V(y^2 - x^2(x+1))$, apply the Jacobian criterion at $(0,0)$: $(\partial f/\partial x)(0,0) = 0$ and $(\partial f/\partial y)(0,0) = 0$, so $T_{X,(0,0)} = \mathbb{A}^2$ has dimension 2 while $\dim X = 1$. Then pick any smooth point, e.g., $(1, \sqrt{2})$: $\partial f/\partial y = 2y \neq 0$, so the Jacobian has rank 1 and $\dim T_{X,P} = 1 = \dim X$.

Week 8 — Local Structure of Morphisms

Concepts to understand:

Local ring map induced by $f: X \to Y$ at a point: $\mathcal{O}_{Y, f(P)} \to \mathcal{O}_{X,P}$
Finite morphisms: $k[Y] \to k[X]$ makes $k[X]$ a finite $k[Y]$-module
Ramification: when a finite map to a smooth curve is not étale at a point
Fiber dimension theorem revisited: generic smoothness and structure of generic fibers
Chevalley’s theorem: the image of a constructible set is constructible

Reading:

Shafarevich Vol 1, §II.3 (~3 hrs)

Problems:

Shafarevich II.3 #1, 2, 3, 4

Milestone

Show that $f: \mathbb{A}^1 \to V(y^2 - x^3),\ t \mapsto (t^2, t^3)$ is a bijection on points but not an isomorphism of varieties. The coordinate ring map $f^*: k[x,y]/(y^2 - x^3) \to k[t]$ sends $x \mapsto t^2, y \mapsto t^3$, but $t \notin \text{Im}(f^*)$ — so $f^*$ is not surjective, $f$ has no algebraic inverse, and the cusp is not isomorphic to $\mathbb{A}^1$ despite being homeomorphic (in the Zariski topology).

Week 9 — Normalization

CA prerequisite: A&M Ch 5 (integral dependence, Noether normalization) — read this week.

Concepts to understand:

Normal domain: integrally closed in its fraction field
Normalization $\tilde{X}$ of $X$: variety corresponding to the integral closure of $k[X]$ in $k(X)$
Normalization is a birational finite morphism $\nu: \tilde{X} \to X$
Noether normalization: every affine variety of dimension $d$ admits a finite surjective map to $\mathbb{A}^d$
A curve is normal iff it is smooth (in dimension 1: regular = normal)

Reading:

Shafarevich Vol 1, §II.4 (~2.5 hrs)
A&M Ch 5 (~2.5 hrs)

Problems:

Shafarevich II.4 #1, 2, 3
A&M 5.1, 5.4, 5.16, 5.22
Compute the normalization of the cuspidal cubic $V(y^2 - x^3) \subset \mathbb{A}^2$

Milestone

Compute the normalization of $V(y^2 - x^3)$ explicitly: set $t = y/x \in k(X)$. Then $t^2 = y^2/x^2 = x^3/x^2 = x$, so $x = t^2$ and $y = t^3$ are in $k[t]$. This shows $k[t] = $ integral closure of $k[t^2, t^3]$ in $k(X)$, and the normalization map is $\mathbb{A}^1 \to V(y^2 - x^3),\ t \mapsto (t^2, t^3)$ — the same map from Week 8, now understood as the normalization.

Week 10 — Resolution of Curve Singularities

Math 137 Lec 24 introduced blowing up; this week goes further to resolution.

Concepts to understand:

Blowing up revisited: strict transform of a curve $C$ under blowing up at $P$
Resolution of the node $V(y^2 - x^2(x+1))$ and the cusp $V(y^2 - x^3)$ by successive blowing up
Every curve over a perfect field has a smooth projective model (resolution in dimension 1)
The smooth projective model is unique up to isomorphism

Reading:

Shafarevich Vol 1, §II.4–II.5 (~3 hrs)
Reid, Ch 7 §7.1–7.2 (~1 hr)

Problems:

Shafarevich II.5 #1, 2, 3, 4
Resolve the $A_2$ singularity $V(y^2 - x^3)$ by two blow-ups. Describe the exceptional locus.

Milestone

Resolve the cusp $V(y^2 - x^3)$ by a single blow-up at the origin. In the chart $y = tx$: total transform is $t^2 x^2 = x^3$, factoring as $x^2(t^2 - x) = 0$. The strict transform is $V(t^2 - x)$, which is smooth and isomorphic to $\mathbb{A}^1$ via the coordinate $t$. It meets the exceptional divisor $E = V(x)$ at the single point $(t, x) = (0, 0)$, confirming the cusp is resolved in one step.

Week 11 — Divisors on Curves

Math 137 Lec 11 covered DVRs and valuations. This week builds the divisor theory on top of that.

Concepts to understand:

Divisor on a smooth projective curve $X$: $D = \sum_{P \in X} n_P [P]$ with finitely many nonzero $n_P$
Degree of a divisor: $\deg D = \sum n_P$
Principal divisor $(f)$ of $f \in k(X)^\times$: zeros minus poles with multiplicity (using the DVR valuation)
Divisor class group (Picard group) $\text{Pic}(X) = \text{Div}(X)/\text{PDiv}(X)$
$\text{Pic}^0(X)$ = degree-0 part; $\text{Pic}(X) \cong \mathbb{Z} \oplus \text{Pic}^0(X)$ for connected curves

Reading:

Shafarevich Vol 1, §III.1 (~3 hrs)

Problems:

Shafarevich III.1 #1, 2, 3, 4, 5
Show $\text{Pic}(\mathbb{A}^1) = 0$ and $\text{Pic}(\mathbb{P}^1) \cong \mathbb{Z}$

Milestone

Compute $\text{div}(f)$ for $f = (x - a)/(x - b) \in k(\mathbb{P}^1)$ in affine coordinate $x$: $\text{div}(f) = [a:1] - [b:1]$. This shows every degree-0 divisor on $\mathbb{P}^1$ is principal, so $\text{Pic}^0(\mathbb{P}^1) = 0$. Since $\text{Pic}(\mathbb{P}^1) \cong \mathbb{Z} \oplus \text{Pic}^0(\mathbb{P}^1)$, conclude $\text{Pic}(\mathbb{P}^1) \cong \mathbb{Z}$, generated by the class of any point.

Week 12 — Linear Systems and Maps to Projective Space

Concepts to understand:

Linear system $|D|$: the projective space of effective divisors linearly equivalent to $D$
Complete linear system and the map $\phi_D: X \dashrightarrow \mathbb{P}(H^0(X, \mathcal{O}(D)))^\vee$
Base locus: points where all sections vanish; $\phi_D$ is a morphism iff base locus is empty
A divisor $D$ is very ample iff $\phi_D$ is a closed immersion
Connection to Math 137 Lec 19 (linear systems of plane curves): the same projective space, now intrinsically defined

Reading:

Shafarevich Vol 1, §III.1 (continued) (~2 hrs)
Reid, Ch 9 §9.1–9.3 (~2 hrs)

Problems:

Show that $\mathcal{O}(1)$ on $\mathbb{P}^1$ corresponds to the identity map $\mathbb{P}^1 \to \mathbb{P}^1$
Find all effective divisors linearly equivalent to $2[P]$ on a smooth conic $C \subset \mathbb{P}^2$

Milestone

On $\mathbb{P}^1$, describe the complete linear system $|n[P]|$ for any point $P$: it is the set of all effective divisors of degree $n$, parameterized by $\mathbb{P}^n$ (via the $n+1$ monomials of degree $n$ in the homogeneous coordinates). Check $n = 1$: $|[P]| \cong \mathbb{P}^1$ and $\phi_{[P]}$ is the identity map on $\mathbb{P}^1$. Check $n = 2$: $|2[P]| \cong \mathbb{P}^2$ and $\phi_{2[P]}$ is the degree-2 Veronese $\mathbb{P}^1 \hookrightarrow \mathbb{P}^2$.

Week 13 — Differential Forms on Curves

Concepts to understand:

Module of Kähler differentials $\Omega_{k[X]/k}$; the sheaf $\Omega_{X/k}$
For a smooth curve: $\Omega_{X/k}$ is a line bundle, the canonical bundle $\omega_X$
Divisor of a differential form: $(\omega) = \sum v_P(\omega) [P]$ via the DVR valuation at each point
Canonical class $K_X$: the divisor class of any nonzero differential form
On $\mathbb{P}^1$: $K_{\mathbb{P}^1} \sim -2[P]$, so $\deg K_{\mathbb{P}^1} = -2$

Reading:

Shafarevich Vol 1, §III.3 (~3 hrs)

Problems:

Shafarevich III.3 #1, 2, 3
On $\mathbb{P}^1$ with coordinate $t$, compute $\text{div}(dt)$. What is $\deg K_{\mathbb{P}^1}$?

Milestone

Compute $\text{div}(dt)$ on $\mathbb{P}^1$ explicitly: in the affine chart $t$, the form $dt$ is regular and nonvanishing. In the chart $s = 1/t$ near $\infty$, $dt = -ds/s^2$ has a double pole at $s = 0$. So $\text{div}(dt) = -2[\infty]$ and $\deg K_{\mathbb{P}^1} = -2 = 2(0) - 2$, confirming the formula $\deg K = 2g - 2$ for $g = 0$.

Week 14 — Genus and the Riemann-Roch Theorem

Concepts to understand:

Geometric genus $g$: $g = \dim H^0(X, \omega_X)$ (holomorphic differentials)
Riemann-Roch theorem: $\ell(D) - \ell(K - D) = \deg D + 1 - g$
Special cases: $\ell(K) = g$, $\deg K = 2g - 2$
For $\deg D > 2g - 2$: $\ell(D) = \deg D + 1 - g$ (no correction term)
Genus formula for a smooth plane curve of degree $d$: $g = \binom{d-1}{2}$

Reading:

Shafarevich Vol 1, §III.4 (~3 hrs)
Reid, Ch 9 §9.4–9.7 (~1 hr)

Problems:

Shafarevich III.4 #1, 2, 3, 4
Compute $g$ for a smooth quartic in $\mathbb{P}^2$. Use RR to find $\ell(K)$.
Show: a smooth curve of genus 0 over $\bar{k}$ is isomorphic to $\mathbb{P}^1$

Milestone

For a smooth elliptic curve $E$ (genus 1) and origin $O$, compute $\ell(nO)$ for $n = 0, 1, 2, 3$ by Riemann-Roch: $\ell(0) = 1$, $\ell(O) = 1$, $\ell(2O) = 2$, $\ell(3O) = 3$. The jumps at $n = 2$ and $n = 3$ produce functions $x$ and $y$ with poles only at $O$; the relation $y^2 = x^3 + ax + b$ follows from $\ell(6O) = 6$ and the seven monomials $1, x, y, x^2, xy, y^2, x^3$ being linearly dependent — RR recovers Weierstrass form from scratch.

Week 15 — Hurwitz’s Theorem

Concepts to understand:

A finite morphism of smooth projective curves $f: X \to Y$ of degree $n$
Ramification index $e_P$ at $P$; ramification divisor $R = \sum_P (e_P - 1)[P]$ on $X$
Hurwitz’s theorem: $2g(X) - 2 = n(2g(Y) - 2) + \deg R$ (for separable $f$)
Application: every map $f: X \to \mathbb{P}^1$ of degree $n$ from a genus-$g$ curve has exactly $2g + 2n - 2$ branch points (over $\mathbb{C}$)
Purely inseparable maps in characteristic $p$: $g(X) = g(Y)$

Reading:

Shafarevich Vol 1, §III.4 (Hurwitz) (~2 hrs)
Fulton, Algebraic Curves, Ch 7 (~2 hrs)

Problems:

Deduce the genus formula for a smooth plane curve from Hurwitz by projecting from a point
Harvard qual: “Let’s talk about Riemann-Hurwitz…” — work through the Ogus questions from the problem set

Milestone

For the hyperelliptic map $f: C \to \mathbb{P}^1$ of degree 2 from a genus-2 curve, apply Hurwitz: $2(2) - 2 = 2 \cdot (2(0) - 2) + \deg R$, giving $\deg R = 6$. Since each branch point has $e_P = 2$ (contributing $e_P - 1 = 1$ to $R$), there are exactly 6 branch points. For $C: y^2 = f(x)$ with $\deg f = 5$, these are the 5 finite roots of $f$ plus the point at infinity.

Week 16 — Elliptic Curves: Classical Picture

Concepts to understand:

An elliptic curve is a smooth projective curve of genus 1 with a marked point $O$
Weierstrass form: $y^2 = x^3 + ax + b$ (char $\neq 2, 3$); smoothness iff $\Delta \neq 0$
Group law: $P + Q + R = 0$ iff $P, Q, R$ are collinear; $O$ is the identity
The $j$-invariant $j(E) = 1728 \cdot \frac{4a^3}{4a^3 + 27b^2}$: classifies $E$ over $\bar{k}$ up to isomorphism
Over $\mathbb{C}$: $E \cong \mathbb{C}/\Lambda$ for a lattice $\Lambda$; group law is addition in $\mathbb{C}/\Lambda$

Reading:

Shafarevich Vol 1, §III.3 (elliptic curves), §III.4 (~2 hrs)
Silverman AEC, Ch 1 §1–3 (preview; return in Phase IV) (~2 hrs)

Problems:

Harvard qual: “What are the involutions of an elliptic curve over $\mathbb{C}$?” — work the McMullen questions
Show the group law is associative using Riemann-Roch (sketch): $\text{Pic}^0(E) \cong E$ as sets

Milestone

For $E: y^2 = x(x-1)(x+1)$, the three 2-torsion points are $(0,0), (1,0), (-1,0)$ (where $y = 0$, so each equals its own inverse since $(x,y)^{-1} = (x,-y)$). Verify the group law: the line through $(0,0)$ and $(1,0)$ is $y = 0$, which also passes through $(-1, 0)$, so $(0,0) + (1,0) + (-1, 0) = O$, confirming $E[2] \cong (\mathbb{Z}/2)^2$.

Weeks 17–18 — Hyperelliptic Curves and Phase I Consolidation

Concepts to understand:

Hyperelliptic curve of genus $g$: double cover of $\mathbb{P}^1$ branched at $2g + 2$ points
Canonical map $\phi_K: X \to \mathbb{P}^{g-1}$ (for $g \geq 2$): base-point-free when $X$ is not hyperelliptic
Every genus-2 curve is hyperelliptic; the canonical map is 2:1 onto $\mathbb{P}^1$
Review: the full algebraic $\leftrightarrow$ geometric dictionary built in Phase I
Every classical concept paired with its scheme-theoretic avatar (preview of Phase II)

Reading:

Shafarevich Vol 1, §III.5 (~2 hrs)
Review Phase I notes; read the Phase II translation table (~3 hrs)

Problems:

Show a genus-2 curve is always hyperelliptic using the canonical map
Work 3 more problems from the Harvard qual collection; identify which Phase II concepts they preview

Milestone

For a smooth genus-2 curve $C$, the canonical map $\phi_K: C \to \mathbb{P}^1$ has degree 2 (since $\ell(K) = g = 2$ and $\deg K = 2g-2 = 2$, so the target is $\mathbb{P}^1$). By Hurwitz applied to $\phi_K$: $2(2) - 2 = 2(-2) + \deg R$, so $\deg R = 6$ and $\phi_K$ has exactly 6 ramification points — the Weierstrass points of $C$, where the hyperelliptic involution fixes the curve.

Weeks 19–20 — Phase I Qual Practice

Use these two weeks for intensive qualifying exam problem work on Phase I material.

Week 19: Work Harvard qual problems involving affine/projective varieties, morphisms, and Bézout - [ ] “Is $\mathbb{P}^1 \times \mathbb{P}^1$ a projective variety? Prove it.” — use Segre - [ ] “Find the explicit equation of the image of $\mathbb{P}^1 \times \mathbb{P}^1$ under Segre” - [ ] “Show a hypersurface of degree $d$ has degree $d$” — use Hilbert polynomial - [ ] “Is the twisted cubic the set-theoretic/scheme-theoretic intersection of two surfaces?” — Ogus questions

Week 20: Work Harvard qual problems involving curves, genus, and Riemann-Roch - [ ] “Find the arithmetic genus of $y^3 = x^2 z$” — Frenkel - [ ] “Calculate $H^0(\mathbb{P}^1, \Omega^1)$” — Poonen (preview of cohomology) - [ ] “Describe Weil divisors and Cartier divisors on curves” - [ ] All McMullen elliptic curve questions

Milestone

You should now be able to: (1) define varieties, morphisms, divisors, and the Picard group from scratch; (2) state and apply Riemann-Roch and Hurwitz; (3) compute intersection numbers and apply Bézout; (4) work problems from the Harvard qual involving divisors, Pic, and curves. Open Ritvik’s qual transcript — the question on Hurwitz’s theorem should now be followable end-to-end.