COMPLEX ALGEBRAIC SURFACES CLASS 13

RAVI VAKIL

Contents

1. Fun with rational surfaces 1

Last day we began:

1. Fun with rational surfaces

I’ll start again, but skip over things we did last time.

The surface 𝔽_{0}. This is ℙ^{1}×ℙ^{1}. Its intersection theory is ℤ[h, f]/h^{2} = f^{2} = 0, hf = 1. h and f are the classes of fibers of the two projections to ℙ^{1}. These are traditionally called rulings. Using the divisor h + f, we can embed 𝔽_{0} as a smooth quadric in ℙ^{3}. More precisely than last day: Let ([x; y], [u; v]) be co-ordinates to ℙ^{1} × ℙ^{1}. Then xu, xv, yu, yv are sections of O(h + f), that separate points and tangent vectors, and hence give a closed immersion into ℙ^{3}. If the base field is algebraically closed, all smooth quadrics are isomorphic to ℙ^{1} × ℙ^{1}.

Now, what do the rulings look like in projective space? What’s their degree? Answer: h · (h + f) = 1. They are lines! (Draw pictures of a hyperboloid with one sheet, and show them the lines.) Question: where are the lines in an ellipsoid? Answer: they are complex, so don’t be misled by the real picture.

Many of the other surfaces corresponds to blow-ups of ℙ^{2} at a certain number of points. Before discussing them, here is a useful proposition that I mentioned last day.

Useful proposition. Consider the blow-up of ℙ^{2} at n general points, giving exceptional divisors E_{1}, …, E_{n}. Then the intersection ring on ℙ^{2} is given by

ℤ[H, E_{1}, …, E_{n}]/H^{2} = 1, HE_{i} = 0, E_{i} · E_{j} = 0, E_{i}^{2} = −1.

We can understand divisors and sections of divisors in terms of divisors on ℙ^{2} with certain multiplicities at the E_{j}. More precisely: the vector space of sections of aH − b_{1}E_{1} − ⋯ − b_{n}E_{n} is naturally isomorphic to the vector space of degree a polynomials in ℙ^{2} vanishing with multiplicity at least b_{i} on E_{i}.

Date: Wednesday, November 13.

Sketch of proof. Any divisor in O(aH −b_{1}E_{1} −⋯−b_{n}E_{n}) pushes forward to some divisor D in class aH on ℙ^{2}. The strict transform of that divisor (i.e., somewhat hideously: the closure of its preimage where the blow-up is an isomorphism) is in class aH −(mult_{p_{1}} D)E_{1}− ⋯−(mult_{p_{n}} D)E_{n}. So the actually original divisor must be this, plus some more E_{i}’s, from which b_{i} ≤ mult_{p_{1}} D.

The vector space structure is the same, as both are subvector spaces of the sections over ℙ^{2} − p_{1} − ⋯ − p_{n}.

The surface 𝔽_{1}. As observed before, 𝔽_{1} is (isomorphic to) the blow-up of ℙ^{2} at a point.

Consider the divisor class 2H − E_{1}. This corresponds to conics in ℙ^{2} through p_{1}, which gives a five-dimensional vector space. It separates points and tangent vectors (somewhat obviously away from E_{1}, and not so obviously along E_{1}). Thus we get an immersion of 𝔽_{1} into ℙ^{4}. Its degree is (2H − E) · (2H − E) = 3. We get a cubic surface in ℙ^{4}.

Interpretation as projection from Veronese surface. Recall that we had an embedding of ℙ^{2} into ℙ^{5} via all conics. We can interpret 𝔽_{1} and its cubic embedding in ℙ^{4} as a projection as follows. Suppose 𝔽_{1} = Bl_{[0;0;1]} ℙ^{2}.

ℙ^{2} ↪^{[x_{0}^{2},x_{0}x_{1};x_{1}^{2};x_{0}x_{2};x_{1}x_{2};x_{2}^{2}]}  ℙ^{5}
↑  ↗  ↓ should be dashed
𝔽_{1}  ↪^{[x_{0}^{2},x_{0}x_{1};x_{1}^{2};x_{0}x_{2};x_{1}x_{2}]}  ℙ^{4}.

Project the quartic Veronese surface in ℙ^{5} down into ℙ^{4} from [0; 0; 0; 0; 0; 1]. This is well defined except at that one point of the Veronese surface. We have a rational map ℙ^{2} −−→ ℙ^{4}. By the elimination of indeterminacy theorem, we can resolve the map after some blowups of ℙ^{2}. In fact, we need only one: 𝔽_{1} → ℙ^{2}.

Blow up ℙ^{2} at two points. Now blow up ℙ^{2} at two points. Where are the (−1)-curves? There are obviously two: our two exceptional divisors. But there is one more: the proper transform of the line joining the two points. It has self-intersection number (H − E_{1} − E_{2})^{2} = −1.

When you blow down that “bonus” rational curve, what do you get? In fact: ℙ^{1} × ℙ^{1}. Last time, Eric saw this by interpreting what we just did as an elementary transformation of 𝔽_{1}.

Here’s another way of seeing it. The divisor 2H −E_{1} −E_{1} corresponds to conic through our points, and is very ample. It gives an immersion into ℙ^{3}, and it is degree 2. Get smooth quadric surface. But we know that all smooth quadrics are ℙ^{1} × ℙ^{1}.

What are the maps to the two ℙ^{1}’s? Answer: projection from each of two points. You can see why the proper transform of the line gets sent to a point under these two projections.

Blow up ℙ^{2} at three points, no two on a line. Where are the (−1)-curves? Answer: the 3 exceptional divisors E_{1}, E_{2}, E_{3}, but also the (proper transforms of) the lines through pairs of those points H − E_{i} − E_{j}.

We can make an diagram showing how these six curves intersect, with a vertex for each curve, and an edge for each intersection. In this case, we get a hexagon.

If you blow down those other 3 curves? Answer: ℙ^{2}. The rational map ℙ^{2} −−→ ℙ^{2} is ancient, and is called a Cremona transformation.

Description 1: conics through these 3 points, i.e. 2H − E_{1} − E_{2} − E_{3}. The space of all conics is a six-dimensional vector space, and if we require the conics to vanish at the three points, we knock the vector space down to dimension 3. So this gives a map to ℙ^{2}. It blows down the line H − E_{1} − E_{2}.

Description 2: Consider the rational map ℙ^{2} −−→ ℙ^{2} given by [x; y; z] ↦ [1/x; 1/y; 1/z]. By the elimination of indeterminacy theorem, we can resolve this map after some blowups. In fact, we need 3.

For fun, let’s look more closely. If x = 0 and y, z ≠ 0, then we map to [1; 0; 0]; that’s a line being blown down. If x = 0 and y = 0 and z ≠ 0, it isn’t so clear. In fact, if [x; y; z] = [at, bt, 1] where a and b aren’t both 0, then [x; y; z] ↦ [1/a; 1/b; t]. Then let t go to 0, and we see that the limit is [1/a; 1/b; 0]. Thus the limit depends on the path of approach, and this is a point where the rational map is undefined, and will have to be blown up. That’s p_{i}. And so on...

Aside: What happens if you blow up at 3 points on a line? Get a (-2)-curve.

Blow up ℙ^{2} at four points. Blow up at 4 general points. Get 10 = 4 + \binom{4}{2} (-1)-curves.

Make graph. This is a highly symmetric graph, and is called the Peterson graph. Its automorphism group is S_{5}; both Tyler and Diane gave me arguments for this within minutes of the end of class.

Blow up ℙ^{2} at five points. Blow up at 5 general points. Get 5 + \binom{5}{2} = 15, plus one more: conic. Cubics through these points. Get quartic surface in ℙ^{4}. (It is the complete intersection of two complete quadrics.

The (-1)-curves turn into lines: E_{1} · (3H − ∑ E_{i}) = 1. (H − E_{1} − E_{2}) · (3H − ∑ E_{i}) = 1. (2H − ∑ E_{i}) · (3H − ∑ E_{i}) = 1.

We could have saved ourselves some effort by instead noting that K_{S} = −3H + ∑ E_{i}. The divisor we are using to embed is −K_{S}. If E is a (-1)-curve, then we know that E^{2} = −1, and by the genus formula (K_{S} + E) · E = −2, from which (−K_{S} · E) = 1. Note that we can reverse this: if S is embedded by the anticanonical divisor, then lines correspond to (−1)-curves. Exercise.

Blow up ℙ^{2} at six points. Now this is serious. Blow up at 6 points: Get 6 + \binom{6}{2} + \binom{6}{5} = 6 + 15 + 6 = 27.

Get smooth cubic surface in ℙ^{3}! With 27 lines! We’ll prove (next day) (i) there are no more lines on this surface, (ii) that almost all smooth cubic surfaces are ℙ^{2} blown up at six points and hence have 27 lines, (iii) all smooth cubic surfaces have 27 lines, and (iv) all smooth cubic surfaces are ℙ^{2} blown up at six points.

The automorphism group of the graph of exceptional curves is W (E_{6}), the Weyl group of E_{6}.

Blow up ℙ^{2} at seven, eight, nine points.

Get 56, [I can’t remember the number, maybe 148], ∞ lines. More interesting geometry here too. For example, blow up at 7 points. Get map to ℙ^{2}. It is a double cover. (−1)- curves map to lines. Fact: the branch locus is a quartic plane curve. (−1)-curves map 2-to-1 to the 28 bitangents of a smooth quartic plane curve.

The automorphism group of the graph of exceptional curves is W (E_{n}), for n = 7, 8, 9.