Algebraic surfaces
Bertini’s theorem, Ampleness Criterion, Intersection theory and Riemann-Roch theorem on surfaces
Remark 0.1. Please refer to [Ha, II 7] for ampleness and very ampleness. Another source is Hartshorne’s book: Ample subvarieties of algebraic varieties, Lecture Notes in Mathematics 156. The discussion of intersection can be found in [Ha, V 1], while Riemann-Roch appears in [Ha, appendix A]. The proof of Bertini’s theorem is the one in [G-H, p137]=[Griffiths, Harris]. For more discussion on cone of curves, we refer [Koll´ar, Mori]: Birational geometry of algebraic varieties. In which they give a complete treatment of minimal model program.
Let D be a divisor on X. If D0 ∼ D, then O(D) ∼= O(D0). Thus sometime it’s useful to pick a better element in |D| instead of looking at D itself.
Theorem 0.2 (Bertini). Let Bs|D| be the base locus of |D|. If dim|D| ≥ 1, then the general member of |D| is non-singular away from the Bs|D|.
In particular, if |D| is base point free, then general member in |D| is non-singular.
Proof. Fix D0, D1∈ |D| with local equation f, f + g on an affine open set U ⊂ X.
We have a subseries (a pencil) Dλ ∈ |D| locally defined by f + λg. Suppose that Pλ∈ U is a singular point of Dλnot in the base locus B := Bs|D|. We may assume that g(Pλ) 6= 0. We have
f + λg(Pλ) = 0,
and ∂
∂zi
(f + λg)(Pλ) = 0, for all zi. Where z1, ..., zn are the local coordinates.
These equations defines a subvariety in Z ⊂ U × P1. And let V = pr1(Z) ⊂ U . One note that f /g is locally constant (−λ) on V − B. Hence for λ different from value of f /g, Dλis non-singular away from B (in U ).
Next one notice that one can cover X by finitely many affine open sets. ¤ Remark 0.3. If |D| is very ample, then it is base point free.
Definition 0.4. A divisor D is said to be ample if mD is very ample for some m > 0.
Our first aim is the following:
Theorem 0.5. Let D be a divisor on a projective variety X. There exist a very ample divisor A such that A + D is very ample.
Corollary 0.6. Let D be a divisor on a projective variety X. Then there are non-singular very ample divisor Y1, Y2 such that D ∼ Y1− Y2
Lemma 0.7. The following are equivalent:
(1) D is ample.
(2) For every coherent sheaf F on X, we have Hi(X, F⊗O(nD)) = 0 for all i > 0 and n À 0.
(3) For every coherent sheaf F on X, F⊗O(nD) is globally generated for all n À 0.
Remark 0.8. A sheaf is globally generated if the natural map H0(X, F)⊗OX→ F is surjective. If F = O(D) for some divisor D, then O(D) is globally generated if and only if D is base point free.
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Exercise 0.9. Show that if D1is very ample and D2is base point free, then D1+D2
is very ample.
(Hint: consider the subspace L(D1)⊗L(D2) ⊂ L(D1+ D2). Show that the map defined by this subspace is everywhere defined and an embedding. Thus the map defined by D1+ D2 is an embedding.
proof of theorem 0.5. X is projective, then X ,→ Pn for some n. Take H a hyper- plane in Pn, then H ∩ X is a very ample divisor on X. By abuse the notation, we still called it,denoted H ,a hyperplane section.
Note that very ample is clearly ample. Hence by the Lemma 0.8 (3), there is an n0 such that D + n0H is base point free. By the exercise, D + (n0+ 1)H is very
ample. ¤
We are now able to define intersection of subvarieties. We start by considering intersection on surface.
Theorem 0.10. Let X be a non-singular projective surface. There is a unique pairing Div(X) × Div(X) → Z, denoted by C.D for any two divisor C, D, such that
(1) if C and D are non-singular curves meeting transversally, then C.D =
#(C ∩ D),
(2) it is symmetric. i.e. C.D = D.C,
(3) it is additive. i.e. (C1+ C2).D = C1.D + C2.D,
(4) it depends only on the linear equivalence classes. i.e. if C1 ∼ C2 then C1.D = C2.D.
Proof. See [Ha, V 1.1]. ¤
Remark 0.11. Let X be a projective variety. An 1-cycle is a formal linear com- bination of irreducible curves. The group of all 1-cycles is denoted Z1(X) (=free abelian group on irreducible curves). One can similarly define a pairing Z1(X) × Div(X) → Z.
Two curves C1, C2 are said to be numerically equivalent if C1.D = C2.D for all D, denoted C1≡ C2. We define
N1(X) := Z1(X)⊗R/ ≡ .
It’s a famous theorem asserts that N1(X) is finite dimensional. Its dimensional is called Picard number, denoted ρ(X).
Remark 0.12. Let V ⊂ X be a subvariety of codimension i, and D is a divisor.
Then it make sense to consider V.Diby decomposing D ∼ H1−H2and then compute (V ∩ Hi).Di−1 in V ∩ Hi inductively on dimension. One can simply set
V.Di:= (V ∩ H1).Di−1− (V ∩ H2).Di−1.
Remark 0.13. Let X be a variety over C. A divisor D gives a class c1(D) ∈ H2(X, Z) via Div(X) → H1(X, O∗) → H2(X, Z). And a curve C give rise to a class [C] ∈ H2(X, Z). The pairing H2(X, Z) × H2(X, Z) → Z gives an intersection theory.
An important feature of ampleness is that it’s indeed a ”numerical property”.
Theorem 0.14 (Nakai’s criterion). Let X be a projective variety. A divisor D is ample if and only V.Di > 0 for all subvariety of codimension i.
In particular, if dimX = 2, then D is ample if and only if D.D > 0 and D.C > 0 for all irreducible curve C.
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Another important criterion is due to Kleiman. Let N E(X) ⊂ N1(X) be the cone generated by effective curves. And let N E(X) be its closure.
For any divisor D, it defines a linear functional on N1(X) and we set D>0 = {x ∈ N1(X)|(x.D) > 0}.
Theorem 0.15 (Kleiman’s criterion). D is ample if and only if D>0⊃ N E(X) − {0}.
Before we revisit the Riemann-Roch theorem on surface, we need the useful adjunction formula:
Proposition 0.16 (Adjunction formula). Let S ⊂ X be a non-singular subvariety of codimension 1 in a non-singular variety X. Then KS := KX+S|S. In particular, if dimX = 2 then 2g(S) − 2 = (KX+ S).S
Given a codimension 1 subvariety Y ⊂ X and a divisor D ∈ Div(X) . One can consider the restriction D|Y, which is supposedly to be a divisor. However, this is not totally trivial. For D = P
niDi, one might want to consider naively that D|Y :=P
ni(Di∩ Y ). But what if Di = Y for some i? That is, how to define Y |Y? One way to think of this is that we deform Y such that limt→0Yt= Y , then we take Y |Y := limt→0Yt|Y. (This needs some extra care).
proof of adjunction formula. Recall that a canonical divisor is a divisor defined by n-forms if dimX = n. Thus one has ΩnX ∼= OX(KX). Where ΩnX denote the sheaf of n-forms on X. Also one can consider sheaf of n-forms on X with pole along S, denoted ΩnX(S). It’s clear that ΩnX(S) ∼= OX(KX+ S).
One has the following exact sequence
0 → OX(KX) → OX(KX+ S) → OS(KX+ S|S) → 0.
On the other hand, one has the Poincar´e residue map ΩnX(S) ³ Ωn−1S
with kernel ΩnX. Comparing these two sequences, one sees that Ωn−1S ∼= OS(KX+ S|S). Hence the canonical divisor KS = KX+ S|S.
We now describe the Poincar´e residue map. (cf. [G-H, p147]). The problem is local in nature, it suffices to describe it locally. We may assume that on a small open set U , S is defined by f . And let z1, .., zn be the local coordinates of U .
The sheaf ΩnX(S) on U can be written as ω = g(z)dzf (z)1∧...∧dn. Since S is non- singular, then at least one of ∂z∂f
i 6= 0. The residue map send ω to ω0:= (−1)i−1g(z)dz1∧ ... ∧ cdzi∧ ... ∧ dn
∂f /∂zi |f =0. This is independent of choice of i since on S
df = ∂f
∂z1
dz1+ ... + ∂f
∂zn
dzn= 0.
Another way to put it is that the residue map sends ω to ω0such that ω = dff∧ω0. It’s clear that the ω0 = 0 if and only if f (z)|g(z), which means that ω is indeed in ΩnX.
¤ Theorem 0.17 (Riemann-Roch theorem for divisors on surfaces). Let X be a non- singular projective surface and D ∈ Div(X) a divisor on X, then one has
χ(X, D) = χ(X, OX) +1
2D.(D − KX).
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Proof. Write D ∼ H1 − H2 with Hi are non-singular very ample divisor. We consider the sequences:
0 → O(D) ∼= O(H1− H2) → O(H1) → OH2(H1) → 0, 0 → O → O(H1) → OH1(H1) → 0.
It’s clear that
χ(X, D) = χ(X, H1) − χ(H2, OH2(H1))
= χ(X, OX) + χ(H1, OH1(H1)) − χ(H2, OH2(H1)).
By Riemann-Roch on curves and adjunction formula, χ(H1, OH1(H1)) = H1.H1+ 1 − g(H1) = H1.H1+ 1 − 1
2(KX+ H1).H1, χ(H2, OH2(H1)) = H1.H2+ 1 − 1
2(KX+ H2).H2. Collecting terms, one has
χ(X, D) = χ(X, OX) +1
2(H1− H2).(H1− H2− KX) = χ(X, OX) +1
2D.(D − KX).
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