Algebraic surfaces

Algebraic surfaces

Ruled surfaces

Remark 0.1. This is a combination of [Be] and [Ha].

Definition 0.2. A vector bundle of rank r over X is π : E → X such that π is locally trivial and patching together by regular functions.

That is, there is an open covering U = {Ui} of X and trivialization maps ϕi : π⁻¹(Ui) → Ui × A^r such that the transition function ϕij : U_i∩ U_j → GL(r, k) is regular.

Note that ϕ_ij is defined as

ϕ_i◦ ϕ⁻¹_j (x, v) = (x, ϕ_ij(x)v).

Definition 0.3. Let π1 : E1 → X and π2 : E2 → X be vector bundles.

An morphism of vector bundles is a morphism f : E1 → E2 such that π2 ◦ f = π1.

Exercise 0.4. Show that

H¹(X, GL(r, k)) ∼= {isomorphic classes of vector bundles of rank r over X}.

A rank 1 vector bundle is call a line bundle. It’s clear that for a given line bundle D, one can associate a line bundle L(D) with transition function f_i/f_j, where f_i are the local defining equations of D. One can prove that

Exercise 0.5. Two divisor D1, D2 are linearly equivalent if and only if L(D1), L(D2) are isomorphic line bundles.

Therefore, there is a injective group homomorphism (Div(X)/ ∼) → H¹(X, GL(1, k)) = H¹(X, O_X^∗).

It’s in fact an isomorphism . We hence call it the Picard group of X and denoted it by Pic(X).

Remark 0.6. The line bundle L(D) induces the sheaf of sections nat- urally. It’s easy to that the sheaf is actually OX(D).

One can similarly define a P^r-bundle over X as we did for vector bundle. The only difference is that the transition function has value in P GL(r + 1, k) = Aut(P^r).

Example 0.7. A P¹-bundle over a curve B is a ruled surface.

Recall that by a (geometrically) ruled surface over B, we mean a surface X with a smooth morphism π : X → B such that each fiber is isomorphic to P¹. Indeed, by the following theorem, one can see that a ruled surface is a P¹-bundle.

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Theorem 0.8 (Noether-Enriques). Let X be a surface and π : X → B be a morphism to a curve B. If x ∈ B is a point such that π is smooth over x and π⁻¹(x) ∼= P¹, then there is an Zariski open set U 3 x such that π⁻¹(U) ∼= U × P¹.

One observe that there is a exact sequence of groups 1 → k^∗ → GL(2, k) → P GL(2, k) → 1.

Over a variety X , then we have an exact sequence of sheaves:

1 → O_X^∗ → GL(2, k) → P GL(2, k) → 1.

As we can identify ruled surface over B with P¹-bundle over B , then the following induced exact sequence is important and illuminating for studying ruled surface.

H¹(B, O^∗_B) → H¹(B, GL(2, k)) → H¹(B, P GL(2, k)) → H²(B, O^∗_B).

Since dimB = 1, one has H²(B, O^∗_B) = 0. One can conclude that (1) Every P¹-bundle over a curve B is P(E) for some rank 2 vector

bundle E over B.

(2) P(E₁) ∼= P(E2) if and only if E₁ ∼= E2⊗L for some line bundle L.

It’s therefore essential and convenient to study rank 2 vector bundle over a curve B.

Let π : E → B be a rank 2 vector bundle. By replacing E with E⊗Lⁿfor some ample line bundle L, we may assume that h⁰(B, E) > 0.

Thus one has 0 → OB → E. (We abuse the notation of sheaves of line bundles(divisors) and line bundles). Let Q be the quotient. It’s clear that Q is a sheaf generically of rank 1. Take Q^∨∨, it’s then a line bundle. Projectivize E → Q^∨∨ → 0, one has

σ : X ∼= P(Q) → P(E).

This is called a section of the ruled surface π : P(E) → X. It’s clear that π ◦ σ = id_X. However, this depends on choice of n.

Another way around this is via the tautological bundle. Let π : P(E) → B be a ruled surface. One has naturally on P(E)

0 → N → π^∗E → Q → 0.

Q is called the tautological line bundle and denoted by O_P(E)(1). Q is associated to a divisor C such that C.F = 1.

Proposition 0.9. Let π : X = P(E) → B be the P¹-bundle as before.

We keep the notation as above. We have (1) PicX = π^∗PicB ⊕ Z[C].

(2) Num(X) = Z[C] ⊕ Z[F ].

(3) C² = deg(E). Recall that deg(E) := deg(∧²E).

(4) K_X ∼ −2C + (deg(E)F + π^∗K_B).

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Proof. (1) For a given divisor D ∈ Div(X), let m = D.F . We consider D⁰ = D − mC, it’s clear that D⁰.F = 0. We claim that D⁰ is linearly equivalent to π^∗G for some G ∈ Div(B).

To this end, we consider D_n := D⁰ + nF and let n go to infinity. Note that D_n² = D², F.K_X = −2, and hence D_n.K_X = D.K_X − 2n. By Riemann-Roch,

χ(X, O(D_n)) = χ(X, O_X) + 1

2(D²− D.K_X + 2n) > 0, ∀n À 0.

On the other hand, h²(X, O(Dn) = h⁰(X, O(KX− Dn)) = 0 for n À 0. Thus, Dn is effective for n À 0. Pick any ∆ ∈ |Dn|, one sees that ∆.F = 0 and hence π(∆) 6= B. If follows that

∆ = π^∗G⁰ for some G⁰ ∈ Div(B). And therefore D ∼ ∆−nF = π^∗(G⁰ − nπ(F )).

(2) By (1), one has NumX = Z[C]⊕π^∗NumB. Since deg : NumB → Z is an isomorphism. We are done.

(3) This is more subtle which we will treat later. We need some thing about the Chern classes in order to prove this. For basic notions, please see the appendix following the proof.

We consider the exact sequence 0 → N → π^∗E → Q → 0, on X.

One has

π^∗c₁(∧²E) = π^∗c₁(E) = c₁(π^∗E) = c₁(N) + c₁(Q).

And

c₁(N).c₁(Q) = c₂(π^∗E) = π^∗c₂(E) = 0 because c₂(E) ∈ H⁴(B, Z) = 0. It follows that

C² = c₁(Q).c₁(Q) = c₁(Q).c₁(π^∗∧²E) − c₁(Q).c₁(N).

Since Q = L(C) and ∧²E = L(∆) for some divisor ∆ of degree deg(E). We have C² = C.π^∗∆ = deg(E).

(4) Since K_X.F = −2, by (1) one has that K_X ∼ −2C + π^∗E for some E. In NumX, [K_X] = −2[C]+b[F ]. And let Γ be a section of π : X → B, then [Γ] = [C] + r[F ]. By adjunction formula,

2g(B) − 2 = 2g(Γ) − 2 = −deg(E) + b.

Fix a point P₀ ∈ B, and fix F₀ to be the fiber over P₀ then (K_X + 2C − π^∗K_B− deg(E)π^∗B₀).F = 0. By (1),

K_X ∼ −2C + π^∗K_B+ deg(E)π^∗B₀+ π^∗∆, for some ∆ with degree = 0.

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Let X be a non-singular projective variety of dimension n and let E be a vector bundle of rank r on X. One can define Chern classes c_i(E) ∈ H²ⁱ(X, Z) for i = 0, .., r. Moreover, one can have total Chern class

c₀(E) + c₁(E) + c₂(E) + ... + c_r(E) and Chern polynomial

c_t(E) := c₀(E) + c₁(E)t + c₂(E)t²+ ... + c_r(E)t^r satisfying:

(1) If E = L(D) is a line bundle induced from a divisor D, then ct(E) = 1+[D]t, where [D] denote the image of D in Div(X) → H¹(X, O_X^∗) → H²(X, Z).

(2) If f : Y → X is a morphism of non-singular varieties, then c_i(π^∗E) = π^∗(c_i(E)).

(3) If 0 → E₁ → E → E₂ → 0 is an exact sequence of vector bundles, then

c_t(E) = c_t(E₁)c_t(E₂),

where the product of classes are the cup product.