04.05.E1 · algebraic-geometry / surfaces

Surfaces exercise pack (Hartshorne Ch. V supplement)

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Formal definition of the pack Intermediate

Hartshorne Chapter V develops the theory of smooth projective surfaces over an algebraically closed field $k$ through one organising structure: the intersection pairing $Pic (X) \times Pic (X) \to Z$ 04.05.06. From it flow the adjunction formula 04.05.07, surface Riemann-Roch 04.05.08, and the Hodge index theorem 04.05.09; the geometric content is carried by blow-ups 04.07.02, ruled and rational surfaces, and the showcase example of the cubic surface with its 27 lines. The chapter's exercises continually combine these — an intersection count feeds an adjunction genus, which feeds a Riemann-Roch dimension, which constrains a linear system.

This pack collects nine exercises — two easy, four medium, three hard — each with a hint and a full solution. It is meant to be read alongside its prerequisite units rather than as a standalone development. The exercises group loosely by Hartshorne section: intersection-number and self-intersection warm-ups (easy), adjunction and blow-up computations (medium), and Nakai-Moishezon, Hodge-index, and cubic-surface arguments (hard).

The conventions throughout are Hartshorne's: $X$ is a smooth projective surface; $D \cdot D^{'}$ the intersection number; $K_{X}$ the canonical divisor; for an irreducible curve $C \subset X$ , adjunction reads $2 p_{a} (C) - 2 = C \cdot (C + K_{X})$ with $p_{a}$ the arithmetic genus. Blowing up a point $P$ produces $π : X \to X$ with exceptional curve $E ≅ P^{1}$ , $E^{2} = - 1$ , and $K_{X} = π^{*} K_{X} + E$ .

Key theorem with full solution Intermediate

Before the pack proper, we work one exercise in full as an exemplar of the format. The remaining eight follow the same structure (problem, hint, full answer in <details> blocks).

Lead exercise. State and prove the Nakai-Moishezon criterion's content on a surface: a divisor $D$ on a smooth projective surface $X$ is ample iff $D^{2} > 0$ and $D \cdot C > 0$ for every irreducible curve $C \subset X$ .

Solution. Ampleness means some multiple $n D$ is very ample, embedding $X$ in projective space 04.05.05. We prove the surface case of Nakai-Moishezon (Hartshorne V.1.10).

Necessity. If $D$ is ample, embed $X ↪ P^{N}$ by $∣ n D ∣$ . Then $n D$ is a hyperplane class $H$ ; $H \cdot C = de g C > 0$ for every curve $C$ (a curve has positive degree under a projective embedding), and $H^{2} = de g X > 0$ (the surface has positive degree). Dividing by $n^{2}, n$ gives $D^{2} > 0$ and $D \cdot C > 0$ .

Sufficiency. Suppose $D^{2} > 0$ and $D \cdot C > 0$ for all irreducible $C$ . By surface Riemann-Roch 04.05.08, $χ (O_{X} (n D)) = χ (O_{X}) + \frac{1}{2} n D \cdot (n D - K_{X})$ grows like $\frac{1}{2} n^{2} D^{2} \to + \infty$ . Serre duality bounds $h^{2} (n D) = h^{0} (K_{X} - n D)$ , which vanishes for $n ≫ 0$ since $(K_{X} - n D) \cdot D < 0$ rules out effectivity against the positive $D$ . Hence $h^{0} (n D) \geq χ (n D) \to \infty$ , so $∣ n D ∣$ is nonempty and grows; a positivity-against-every-curve argument (Nakai's induction on subvarieties) upgrades this to base-point-freeness and then very-ampleness of a large multiple. Therefore $D$ is ample. $□$

This is the Nakai-Moishezon criterion specialised to surfaces — the numerical characterisation of ampleness. It turns "ample" (a hard geometric condition about embeddings) into two intersection inequalities, computable from the Néron-Severi lattice 04.05.02. Most ampleness checks downstream reduce to it.

Exercises Intermediate

Exercise 1 (easy, numeric). Intersection numbers on $P^{1} \times P^{1}$ .

On $Q = P^{1} \times P^{1}$ let $f, g$ be the two ruling classes (fibers of the two projections). Compute $f^{2}$ , $g^{2}$ , $f \cdot g$ , and the self-intersection of the diagonal $Δ$ , whose class is $f + g$ .

Hint

Two distinct fibers of the same ruling are disjoint, so $f^{2} = g^{2} = 0$ ; one fiber of each ruling meets in a single point, so $f \cdot g = 1$ 04.05.06.

Answer

Fibers of one projection are disjoint (move one fiber to another value of the parameter), so $f^{2} = 0$ and $g^{2} = 0$ . A fiber of the first projection and a fiber of the second meet in exactly one point transversally, so $f \cdot g = 1$ .

The diagonal $Δ \subset Q$ has class $f + g$ (it projects isomorphically to each factor, meeting each ruling once). Its self-intersection:

Δ^{2} = (f + g)^{2} = f^{2} + 2 f \cdot g + g^{2} = 0 + 2 \cdot 1 + 0 = 2.

Check via adjunction 04.05.07: $Δ ≅ P^{1}$ has $p_{a} = 0$ , and $K_{Q} = - 2 f - 2 g$ , so $2 p_{a} - 2 = Δ \cdot (Δ + K_{Q}) = (f + g) \cdot (- f - g) = - (f + g)^{2} = - 2$ , giving $p_{a} = 0$ . Consistent.

Exercise 2 (easy, numeric). Self-intersection of the exceptional curve.

Let $π : X \to X$ be the blow-up of a smooth surface $X$ at a point $P$ , with exceptional curve $E$ . Compute $E^{2}$ and $E \cdot π^{*} D$ for any divisor $D$ on $X$ .

Hint

$π^{*} D$ is supported away from $E$ generically; $E^{2} = - 1$ is the defining numerical feature of a $(- 1)$ -curve 04.07.02.

Answer

The pullback $π^{*} D$ of a divisor on $X$ meets the exceptional curve in degree zero: $π^{*} D \cdot E = D \cdot π_{*} E = D \cdot 0 = 0$ (the exceptional curve maps to the point $P$ , so $π_{*} E = 0$ ). Hence

E \cdot π^{*} D = 0 for every divisor D on X .

The self-intersection of the exceptional curve is

E^{2} = - 1.

This is the hallmark of the blow-up 04.07.02: $E ≅ P^{1}$ with $E^{2} = - 1$ , a " $(- 1)$ -curve". One way to see it: $O_{X} (E) ∣_{E} ≅ O_{E} (- 1) = O_{P^{1}} (- 1)$ , whose degree is $E \cdot E = - 1$ . The Picard group of $X$ is $π^{*} Pic (X) \oplus Z E$ , orthogonal with $E^{2} = - 1$ .

Exercise 3 (medium, numeric). Genus of a curve through a blown-up point.

Let $C \subset X$ be a smooth curve passing through $P$ with multiplicity $m$ (an ordinary $m$ -fold point). On the blow-up $π : X \to X$ , the strict transform is $C = π^{*} C - m E$ . Compute $C^{2}$ in terms of $C^{2}$ and $m$ .

Hint

Expand $(π^{*} C - m E)^{2}$ using $π^{*} C \cdot E = 0$ and $E^{2} = - 1$ from Exercise 2.

Answer

Using the blow-up intersection rules from Exercise 2 — $(π^{*} C)^{2} = C^{2}$ , $π^{*} C \cdot E = 0$ , $E^{2} = - 1$ :

C^{2} = (π^{*} C - m E)^{2} = (π^{*} C)^{2} - 2 m (π^{*} C \cdot E) + m^{2} E^{2} = C^{2} - 0 + m^{2} (- 1) = C^{2} - m^{2} .

So passing through $P$ with multiplicity $m$ and taking the strict transform drops the self-intersection by $m^{2}$ :

C^{2} = C^{2} - m^{2} .

This is the standard tool for resolving singularities of plane curves: blowing up a node ( $m = 2$ ) of a plane curve drops $C^{2}$ by $4$ and separates the branches. Combined with $K_{X} = π^{*} K_{X} + E$ and adjunction, it gives the genus-drop formula $p_{a} (C) = p_{a} (C) - (2 m)$ when $C$ is smooth (Hartshorne V.3.7).

Exercise 4 (medium, numeric). Adjunction on the projective plane.

Compute the arithmetic genus of a smooth plane curve $C \subset P^{2}$ of degree $d$ using adjunction, and verify against the genus-degree formula.

Hint

On $P^{2}$ , $K_{P^{2}} = - 3 H$ where $H$ is the line class, $H^{2} = 1$ , and $C \sim d H$ . Apply $2 p_{a} - 2 = C \cdot (C + K)$ 04.05.07.

Answer

On $P^{2}$ : the hyperplane (line) class $H$ has $H^{2} = 1$ , the canonical class is $K_{P^{2}} = - 3 H$ [04.08 canonical sheaf; $ω = O (- 3)$ ], and a degree- $d$ curve has class $C = d H$ . Adjunction 04.05.07 gives

2 p_{a} (C) - 2 = C \cdot (C + K_{P^{2}}) = d H \cdot (d H - 3 H) = d (d - 3) H^{2} = d (d - 3) .

Hence

p_{a} (C) = \frac{d ( d - 3 )}{2} + 1 = \frac{d ^{2} - 3 d + 2}{2} = \frac{( d - 1 ) ( d - 2 )}{2} .

This matches the genus-degree formula derived cohomologically [04.03.E1 Exercise 3]: a smooth plane curve of degree $d$ has genus $(2 d - 1)$ . For $d = 1, 2$ : $p_{a} = 0$ (lines and conics are rational). For $d = 3$ : $p_{a} = 1$ (cubics are elliptic). For $d = 4$ : $p_{a} = 3$ (quartics realise the genus- $3$ canonical curve, Exercise 6 of the curves pack).

Exercise 6 (hard, proof). The Hodge index theorem.

Let $X$ be a smooth projective surface and $H$ an ample divisor. Show that if a divisor $D$ satisfies $D \cdot H = 0$ and $D \neq \equiv 0$ (numerically), then $D^{2} < 0$ . Conclude the intersection form on $NS (X) \otimes R$ has signature $(1, ρ - 1)$ .

Hint

This is the Hodge index theorem 04.05.09. Suppose $D^{2} \geq 0$ ; combine $D$ with $H$ and use Riemann-Roch to produce effective divisors, contradicting $D \cdot H = 0$ .

Answer

Suppose $D \cdot H = 0$ , $D \neq \equiv 0$ , and toward a contradiction $D^{2} \geq 0$ .

Case $D^{2} > 0$ . Consider $D_{n} = D + n H$ for varying $n$ . We have $D_{n}^{2} = D^{2} + 2 n (D \cdot H) + n^{2} H^{2} = D^{2} + n^{2} H^{2} > 0$ . By surface Riemann-Roch 04.05.08 and the growth argument (lead exercise), for $∣ n ∣$ large $D_{n}$ or $- D_{n}$ is effective. But $D_{n} \cdot H = D \cdot H + n H^{2} = n H^{2}$ , which is positive for $n > 0$ and negative for $n < 0$ ; an effective divisor meets the ample $H$ non-negatively, so $D_{n}$ effective forces $n > 0$ and $- D_{n}$ effective forces $n < 0$ — both achievable, yet then $D = \frac{1}{2} (D_{n} + D_{- n} -type)$ relations force $D \cdot H \neq = 0$ unless $D \equiv 0$ . More directly: $D^{2} > 0$ and $D \neq \equiv 0$ make $D$ or $- D$ effective (big), so $D \cdot H \neq = 0$ for ample $H$ — contradicting $D \cdot H = 0$ .

Case $D^{2} = 0$ . Pick any divisor $E$ with $E \cdot D \neq = 0$ (exists since $D \neq \equiv 0$ and the form is non-degenerate on $NS$ ). For real $t$ , consider $D^{'} = t D + E^{'}$ where $E^{'} = E - \frac{E \cdot H}{H ^{2}} H$ satisfies $E^{'} \cdot H = 0$ . Then $(t D + E^{'}) \cdot H = 0$ and $(t D + E^{'})^{2} = 2 t (D \cdot E^{'}) + E^{'2}$ , which is linear in $t$ with nonzero slope $2 (D \cdot E^{'}) \neq = 0$ , so it takes a positive value for some $t$ — reducing to the previous case and yielding a contradiction.

Hence $D^{2} < 0$ . This says: on the hyperplane $H^{⊥} \subset NS (X) \otimes R$ , the intersection form is negative definite. Since $H^{2} > 0$ , the form is positive on the line $R H$ and negative on its orthogonal complement of dimension $ρ - 1$ . The signature is $(1, ρ - 1)$ — the Hodge index theorem (Hartshorne V.1.9). It underlies the structure of the Néron-Severi lattice and the spin-geometry $b_{2}^{+} = 1$ phenomenon for many algebraic surfaces.

Exercise 7 (hard, proof). The cubic surface is $P^{2}$ blown up at six points.

Show that a smooth cubic surface $X \subset P^{3}$ is isomorphic to $P^{2}$ blown up at $6$ points in general position, and use this to count $K_{X}^{2}$ .

Hint

The anticanonical class: by adjunction on $P^{3}$ , $K_{X} = (K_{P^{3}} + X) ∣_{X} = (- 4 H + 3 H) ∣_{X} = - H ∣_{X}$ , so $X$ is anticanonically embedded. Blowing up $6$ general points of $P^{2}$ has $K^{2} = 9 - 6 = 3$ .

Answer

By adjunction for the surface $X = {F_{3} = 0} \subset P^{3}$ 04.05.07:

K_{X} = (K_{P^{3}} + X)_{X} = (- 4 H + 3 H)_{X} = - H_{X},

so $- K_{X}$ is the hyperplane class — $X$ is embedded by its anticanonical system $∣ - K_{X} ∣$ , a del Pezzo surface of degree $K_{X}^{2} = H^{2} ∣_{X} = de g X = 3$ .

Now blow up $6$ points $p_{1}, \dots, p_{6} \in P^{2}$ in general position (no $3$ collinear, not all $6$ on a conic), giving $π : Y \to P^{2}$ with exceptional curves $E_{1}, \dots, E_{6}$ . Then $Pic (Y) = Z ⟨ ℓ, E_{1}, \dots, E_{6} ⟩$ where $ℓ = π^{*} H$ , with $ℓ^{2} = 1$ , $E_{i}^{2} = - 1$ , $ℓ \cdot E_{i} = E_{i} \cdot E_{j} = 0$ ( $i \neq = j$ ). The canonical class is

K_{Y} = - 3 ℓ + E_{1} + \dots + E_{6}, K_{Y}^{2} = 9 \cdot 1 + 6 \cdot (- 1) = 9 - 6 = 3.

The anticanonical system $∣ - K_{Y} ∣ = ∣3 ℓ - \sum E_{i} ∣$ is the system of plane cubics through the $6$ points; for points in general position it is very ample of dimension $ℓ (- K_{Y}) - 1 = 3$ (a net of cubics through $6$ points has $(2 3 + 2) - 6 = 10 - 6 = 4$ dimensions of sections), embedding $Y ↪ P^{3}$ as a cubic surface. Both $X$ and $Y$ are anticanonically embedded del Pezzo surfaces of degree $3$ ; by the uniqueness of the minimal model and the classification (Hartshorne V.4.7), $X ≅ Y$ .

Thus every smooth cubic surface is $P^{2}$ blown up at $6$ general points, with $K_{X}^{2} = 3$ .

Exercise 8 (hard, proof). The 27 lines on a cubic surface.

Using the blow-up model of Exercise 7, enumerate the $27$ lines on a smooth cubic surface and verify the count.

Hint

Lines on $X$ are the $(- 1)$ -curves: classes $C$ with $C^{2} = - 1$ and $C \cdot K_{X} = - 1$ . Sort them by type on the blow-up $P^{2}$ at $6$ points: the $6$ exceptional curves, the $(2 6)$ lines through pairs, the $6$ conics through five points.

Answer

On a cubic surface a line $L$ has $L \cdot (- K_{X}) = L \cdot H = 1$ (lines have degree $1$ ), and by adjunction 04.05.07 $2 p_{a} (L) - 2 = L \cdot (L + K_{X})$ with $p_{a} = 0$ gives $- 2 = L^{2} - 1$ , so $L^{2} = - 1$ . Lines are exactly the $(- 1)$ -curves: $C^{2} = - 1$ , $C \cdot K_{X} = - 1$ .

On $Y = Bl_{6} P^{2}$ with $K_{Y} = - 3 ℓ + \sum E_{i}$ , solve $C = a ℓ - \sum b_{i} E_{i}$ with $C^{2} = a^{2} - \sum b_{i}^{2} = - 1$ and $- C \cdot K_{Y} = 3 a - \sum b_{i} = 1$ . The non-negative solutions (effective irreducible $(- 1)$ -classes) are:

The $6$ exceptional curves $E_{i}$ : class $E_{i}$ , with $E_{i}^{2} = - 1$ . Count: $(1 6) = 6$ .
The $(2 6) = 15$ strict transforms of lines through two of the points: class $ℓ - E_{i} - E_{j}$ , with $(ℓ - E_{i} - E_{j})^{2} = 1 - 1 - 1 = - 1$ . Count: $15$ .
The $6$ strict transforms of conics through five of the six points: class $2 ℓ - \sum_{k \neq = i} E_{k}$ , with $(2 ℓ - \sum_{k \neq = i} E_{k})^{2} = 4 - 5 = - 1$ . Count: $(5 6) = 6$ .

Total:

6 + 15 + 6 = 27.

These are the famous 27 lines (Cayley-Salmon, 1849). Their intersection pattern — each line meets $10$ others — is governed by the root system $E_{6}$ : the $27$ lines correspond to the $27$ weights of the minuscule representation of $E_{6}$ , and the Weyl group $W (E_{6})$ (order $51840$ ) acts as the symmetry group permuting them (Hartshorne V.4.9–4.11). The configuration is the prototype example in the deformation theory of del Pezzo surfaces.

Exercise 9 (medium, proof). Castelnuovo's contractibility — the numerical criterion.

Let $E \subset X$ be a curve on a smooth projective surface with $E ≅ P^{1}$ and $E^{2} = - 1$ . Show that $E \cdot K_{X} = - 1$ , the numerical signature of an exceptional curve of a blow-up.

Hint

Apply adjunction 04.05.07 to $E$ with $p_{a} (E) = 0$ .

Answer

Since $E ≅ P^{1}$ , its arithmetic genus is $p_{a} (E) = 0$ . Adjunction 04.05.07 on the surface $X$ gives

2 p_{a} (E) - 2 = E \cdot (E + K_{X}) = E^{2} + E \cdot K_{X} .

Substituting $p_{a} (E) = 0$ and $E^{2} = - 1$ :

- 2 = - 1 + E \cdot K_{X} ⟹ E \cdot K_{X} = - 1.

So a smooth rational curve with $E^{2} = - 1$ automatically has $E \cdot K_{X} = - 1$ — it is a $(- 1)$ -curve, exactly the numerical type of the exceptional curve of a single blow-up (Exercise 2). Castelnuovo's contractibility criterion (Hartshorne V.5.7) is the converse direction: any such $(- 1)$ -curve can be contracted, i.e. there is a morphism $π : X \to X^{'}$ to a smooth projective surface realising $X$ as the blow-up of $X^{'}$ at a point with $E$ as exceptional curve. This is the engine of the minimal-model program for surfaces: contract $(- 1)$ -curves until none remain, reaching a minimal model. The $27$ lines of Exercise 8 are precisely the $27$ ways to contract a cubic surface back down toward $P^{2}$ .

Exercise pack EP1 for 04-algebraic-geometry/05-surfaces. Hartshorne Chapter V supplement: the intersection pairing, adjunction, Riemann-Roch and Hodge index for surfaces, blow-ups and ruled/rational surfaces, the Nakai-Moishezon criterion, and the cubic surface with its 27 lines (§V.1–§V.4).

Prerequisites

04.05.06 pending
04.05.07 pending
04.05.08 pending
04.05.09 pending
04.05.05 pending
04.07.02 pending
04.05.02 pending

Tier anchors

beginner
intermediate: Hartshorne Ch. V exercises (§V.1 intersection pairing, adjunction, Riemann-Roch for surfaces, Hodge index; §V.2 ruled and rational surfaces; §V.3 monoidal transformations / blow-ups; §V.4 the cubic surface and the 27 lines)
master

References

Hartshorne — Algebraic Geometry (GTM 52) · Ch. V exercises across §V.1 (intersection numbers, adjunction, the Hodge index theorem, Riemann-Roch on a surface), §V.2 (ruled surfaces, the Hirzebruch surfaces $\mathbb{F}_n$, sections and the negative section), §V.3 (blow-up of a point, the exceptional curve $E$ with $E^2=-1$, Castelnuovo's contractibility), §V.4 (the cubic surface as $\mathbb{P}^2$ blown up at 6 points, the 27 lines)
Beauville — Complex Algebraic Surfaces (LMS Student Texts 34) · Ch. I–IV — intersection theory, blow-ups, ruled and rational surfaces, the classification
Reid — Chapters on Algebraic Surfaces · §§A–E — the intersection form, the 27 lines, Castelnuovo's criterion

Estimated time

intermediate: 6h