Surfaces

Report on Meeting 4

Theorem 1. Let X be a smooth surface over a field k. Let π : X̃ → X be the blowup of a k-point P, and let E = π^{−1}(P) be the exceptional curve. Then

(i) The natural maps π^{∗} : P ic(X) → P ic(X̃) and Z → P ic(X̃) (by 1 ↦ E) induce an isomorphism P ic(X̃) = P ic(X) ⊕ Z.

(ii) E^{2} = −1. (iii) If D ∈ P ic(X), then π^{∗}(D).E = 0. (iv) If C, D ∈ P ic(X), then π^{∗}(C).π^{∗}(D) = C.D.

(v) Let K be a canonical divisor on X and let K̃ be a canonical divisor on X̃. Then K̃ = π^{∗}K + E, and so K̃^{2} = K^{2} − 1.

(vi) One has

R^{i}π_{∗}(OX̃) = \{ \begin{matrix}O_{X} & if i = 0, \\ 0 & if i > 0. \end{matrix}

Therefore, H^{i}(X, O_{X}) = H^{i}(X̃, OX̃). (vii) CH(X̃) = CH(X) ⊕ Z. (vii) K_{0}(X̃) = K_{0}(X) ⊕ Z. (viii) The étale cohomology groups satisfy

H_{et}^{i}(X̃, Q_{l}) = \{ \begin{matrix}H^{i}(X, Q_{l}) & if i ≠ 2, \\ H^{i}(X, Q_{l}) ⊕ Z & if i = 2. \end{matrix}

Theorem 2. Every birational isomorphism of surfaces is a composition of blowups of closed points (and their inverses).

Proposition 3. The following invariants of a surface are boring:

• h^{0,0} = h^{2,2} = 1.

• b_{0} = b_{4} = 1.

Proof: Serre duality and trivial computations of zero-dimensional cohomology.

Proposition 4. The following invariants of a surface are birationally invariant: • κ.

• p_{g} := h^{2,0} = h^{0,2}.

• q := dim(Alb_{X}) = h^{1,0} = h^{1,2}.

• s := h^{0,1} = h^{2,1}.

• b_{1} = b_{3} = 2q.

• χ(O_{X}) := 1 − s + p_{g}.

• p_{a} := χ(O_{X}) − 1 = p_{g} − s.

• Δ := 2s − b_{1} = 2(s − q).

Proof: To get the result for the Kodaira dimension, start with a blowup π : X̃ → X, and consider the sheaf OX̃(nE). Now, π : X̃ − E → X − P is an isomorphism, and the sheaf is trivial on this open set. So, we see (i) that there is a map φ : O_{X} → π_{∗}O_{X̃(nE)} that is an isomorphim away from P, and (ii) that R^{i}π_{∗}O_{X̃(nE)} is supported at P. It follows (with some additional work) from the theorem on formal functions that φ is an isomorphism and that the higher direct images vanish. Now one has

π_{∗}(OX̃(nK̃) = π_{∗}(OX̃(nπ^{∗}K + nE)) = π_{∗}OX̃(nE) ⊗ O_{X}(nK) = O_{X}(nK),

where the middle step follows from the projection formula. Thus, the canonical rings are the same, so their transcendence degree is the same.

The next three results follow from the above result on how cohomology changes, as does the first part about the Betti numbers. The fact that b_{1} = 2q follows because the first étale cohomology group classifies unramified abelian coverings, and you can construct them all by pulling back isogenies along the Albanese mapping α : X → Alb_{X}.

Now the remaining results follow by the definitions of the invariants in questions.

Proposition 5. The following invariants of a surface are not birationally invariant: • K^{2}.

• b_{2}.

• h^{1,1}.

• χ_{top} := ∑_{i≥0}(−1)^{i}b_{i} = 2 − 4q + b_{2} = c_{2}.

Proof: By the above computations, K^{2} decreases by one and b_{2} and h^{1,1} both increase by one whenever you blow up a point. The last line defines the topological Euler characteristic χ_{top}, and the expansion of the sum follows from previous computations. Moreover, we know by GHRR that K^{2} + c_{2} = 12χ(O_{X}), which is a birational invariant. Thus, c_{2} also increases by one every time you blow up a point. Since the two sides of the final equality change in the same way, we can replace X by any birationally equivalent surface to verify the equality. Now use the fact that every surface is birationally equivalent to a Lefschetz pencil (a fibration over P^{1}) with irreducible fibres and do the computation of étale cohomology groups on the pencil.

Proposition 6. 0 ≤ Δ ≤ 2p_{g}.

Proof: I only know that you’re supposed to mutter some magic incantationa bout “the Bockstein maps H^{1}(O_{X}) → H^{2}(O_{X})” at this point, and refer the interested reader to Mumford’s book on Curves on an Algebraic Surface.

Proposition 7. One has the following nontrivial relation between the numerical invariants of a surface

10 + 12p_{g} = K^{2} + b_{2} + 8s + 2Δ.

Proof: Start with Noether’s Formula:

12χ(O_{X}) = K^{2} + c_{2}.

Use the relations above to replace χ(O_{X}) and c_{2} and you get

12 − 12s + 12p_{g} = K^{2} + b_{2} − 4q + 2.

Recollect terms and use the definition of Δ = 2(s − q) to get the result.

Remark 8. Here is a table of the constraints on the numerical invariants on surfaces. Notice in particular that the fundamental invariants in the top half of the table are nonnegative, with the possible exception of K^{2}.

Kodaira Dimension  κ  −1 ≤ κ ≤ 2
Geometric Genus  p_{g}  p_{g} ≥ 0
Irregularity  s  s ≥ 0
Δ  even, 0 ≤ Δ ≤ 2p_{g}
Second Betti Number  b_{2}  b_{2} > 0
Canonical Self-Intersection  K^{2}
Albanese Dimension  q  q = s − \frac{1}{2}Δ
First Betti Number  b_{1}  b_{1} = 2q
Euler Characteristic  χ  χ = 1 − s + p_{g}
Arithmetic Genus  p_{a}  p_{a} = p_{g} − s
Second Chern Class  c_{2}  c_{2} = 2 − 4q + b_{2}

In addition to the constraints in the table, one always has

10 + 12p_{g} = K^{2} + b_{2} + 8s + 2Δ,

and Δ = 0 for an algebraic surface in characteristic zero.

Remark 9. The next table show sample values for the numerical invariants. For κ < 1, this table show every possible value on a minimal surface.

\begin{matrix}κ & p_{g} & s & Δ & b_{2} & K^{2} & q & b_{1} & χ & p_{a} & c_{2} \\ BIR & BIR & BIR & BIR & NO(+) & NO(−) & BIR & BIR & BIR & BIR & NO(+) \\ −1 & 0 & 0 & 0 & 1 & 9 & 0 & 0 & 1 & 0 & 3 \\ −1 & 0 & 0 & 0 & 2 & 8 & 0 & 0 & 1 & 0 & 4 \\ −1 & 0 & g & 0 & 2 & 8(1 − g) & g & 2g & 1 − g & −g & 4(1 − g) \\ 0 & 0 & 0 & 0 & 10 & 0 & 0 & 0 & 1 & 0 & 12 \\ 0 & 0 & 1 & 0 & 2 & 0 & 1 & 2 & 0 & −1 & 0 \\ 0 & 1 & 0 & 0 & 22 & 0 & 0 & 0 & 2 & 1 & 24 \\ 0 & 1 & 1 & 0 & 14 & 0 & 1 & 2 & 1 & 0 & 12 \\ 0 & 1 & 2 & 0 & 6 & 0 & 2 & 4 & 0 & −1 & 0 \\ 0 & 1 & 1 & 2 & 10 & 0 & 0 & 0 & 1 & 0 & 12 \\ 0 & 1 & 2 & 2 & 2 & 0 & 1 & 2 & 0 & −1 & 0 \\ 1 & g & g + 1 & 0 & 4g + 2 & 0 & g + 1 & 2g + 2 & 0 & −1 & 0 \end{matrix}