CHAPTER 1

### Prelimanary

1. An Example

This section is devoted to illustrate the connection between compact Riemann surface and complex algebraic curve. We will basically work out the example of elliptic curves and leave the general discussion for interested reader.

*Let Λ ⊂ C be a lattice, that is, Λ = Zλ*_{1}*+Zλ*_{2}*with C = Rλ*_{1}*+Rλ*_{2}.
*Then an elliptic curve E := C/Λ is an abelian group.*

On the other hand, it’s a compact Riemann surface. One would
like to ask whether there are holomorphic function or meromorphic
*functions on E or not.*

*To this end, let π : C → E be the natural map, which is a group ho-*
momorphism (algebra), holomorphic function (complex analysis), and
covering (topology). A function ¯*f on E gives a function f on C which*
*is double periodic, i.e.,*

*f (z + λ*_{1}*) = f (z), f (z + λ*_{2}*) = f (z), ∀z ∈ C.*

Recall that we have the following well-known result:

*Theorem 1.1 (Liouville). A bounded entire function, i.e. a holo-*
*morphic function on C with bounded image, is constant.*

*Corollary 1.2. There is no non-constant holomorphic function*
*on E.*

Proof. First note that if ¯*f is a holomorphic function on E, then*
*it induces a doubly periodic holomorphic function f on C such that*
*f ◦π = f . It then suffices to claim that a a doubly periodic holomorphic*¯
function on C is bounded.

*To do this, consider D := {z = α*_{1}*λ*_{1} *+ α*_{2}*λ*_{2}*|0 ≤ α*_{i}*≤ 1}. The*
*image of f is f (D). However, D is compact, so is f (D) and hence f (D)*
*is bounded. By Liouville theorem, f is constant and hence so is ¯f . ¤*
*The next hope is to ask if there is a meromorphic function on E*
with only a simple pole or not. The answer is NO, which can be proved
by residue theorem.

*Exercise 1.3. Prove that there is no non-zero meromorphic func-*
*tion on E with only a simple pole*

1

2 1. PRELIMANARY

*Hint. If the pole is not on the boundary of D, then consider the*
*path integral along the boundary of D, otherwise consider the path*
*integral along a translation of D. By using Residue theorem, one can*
see it.

Then the next hope is to look for functions with pole of order 2.

*Luckily, we have one, which is the Weierstrass P-function,*
*P(z) := z** ^{−2}*+ X

*ω∈Λ−{0}*

*((z − ω)*^{−2}*− ω*^{−2}*).*

*And P** ^{0}* is a functions with pole of order 3. By direct computation, one
sees that

*{wp_equation}*

Lemma 1.4.

*P*^{0}*(z)*^{2} *= 4P(z)*^{3}*− g*_{2}*P(z) − g*_{3}*.*
*Where*

*g*_{2} = 60 X

*ω∈Λ−{0}*

*ω*^{−4}

*and*

*g*_{3} = 140 X

*ω∈Λ−{0}*

*ω*^{−6}*.*

*Exercise 1.5. Verify that P is doubly peiodic, i.e. P(z+λ) = P(z)*
*for all z ∈ C and λ ∈ Λ.*

*Work out the computation in Lemma 1.4.*

*Theorem 1.6. An elliptic curve can be embedded into P*^{2}_{C} *as a non-*
*singular cubic.*

*Sketch. We considering the map ϕ : E − {0} → C*^{2} given by

¯

*z 7→ (P(z), P*^{0}*(z)).*

*This map can be extended to ϕ : E → P*^{2} as

½ *ϕ(¯z) = [P(z), P*^{0}*(z), 1]* if ¯*z 6= 0,*
*ϕ(¯z) = [0, 1, 0]* if ¯*z = 0.*

The affine defining equation in C^{2} is
*y*^{2} *= 4x*^{3}*− g*2*x − g*3*.*
And the projective defining equation is

*y*^{2}*z = 4x*^{3}*− g*2*xz*^{2}*− g*3*z*^{3}*.*

*One can verify that this cubic is non-singular and the map ϕ is an*

embedding. ¤

2. DIVISORS 3

2. Divisors

Similar phenomena occurs for any compact Riemann surface. As the above section suggested, the essential point is to find enough func- tions and then determine the algebraic relation between those func- tions. All these can be done for any compact Riemann surface. Thus the purpose of this section is to show the following

*Theorem 2.1. Any compact Riemann surface can be embedded into*
*a projective space as an algebraic curve.*

*To study functions more systematically, it’s natural to consider di-*
*visors.*

*Definition 2.2. Let X be a compact Riemann surface. A divisor,*
*denoted D =* P

*n*_{i}*P*_{i}*, is a finite formal sum of finite points (codim=1).*

*Given divisors D*1 =P

*n**i**P**i* *and D*2 =P

*m**i**P**i*, one can define
*D*_{1}*+ D*_{2} :=X

*(n*_{i}*+ m*_{i}*)P*_{i}*.*

*Let Div(X) be the set of all divisors on X. It’s clear that Div(X) is a*
free abelian group under the addition defined above. In fact, one can
*think Div(X) as the free abelian group on the set X.*

*Given a meromorphic function f on X, one can count its zeros and*
poles with multiplicity. This give rise to the following group homomor-
phism

*div : M(X) − {0} → Div(X),*

*where M(X) denotes the field of meromorphic functions on X.*

*Exercise 2.3. A meromorphic function on a compact Riemann*
*surface X has at most finitely many zeros and poles.*

*Hence div is well-defined.*

The idea for divisors is to collect information on poles and zeros.

We denote the functions with prescribed poles and zeros, collected in
*D =*P

*n*_{i}*P** _{i}*, as

*L(D) := {f ∈ M(X) − {0}|div(f ) + D ≥ 0}.*

It’s clearly a vector space over the ground field and its dimension is
*denoted l(D). Another important notion for divisor is the degree, which*
is

*deg(D) :=*X
*n*_{i}*.*

*Example 2.4. Let X = C ∪ {∞} be the Riemann sphere. Then*
*M(X) ∼= C(z). We use the notation of [x] to denote the divisor of the*
*point with coordinate z = x.*

*Let f*_{1}*(z) = z, f*_{2}*(z) = 1/(z − 1), f*_{3}*(z) = z/(z − 1)*^{2}*. By easy*
*computation, one finds that div(f*_{1}*) = 1[0] − 1[∞] since it has a zero*

4 1. PRELIMANARY

*at 0 and a pole at ∞. Similarly, div(f*2*) = 1[∞] − 1[1], div(f*3) =
*1[0] + 1[∞] − 2[1].*

*Now fix a divisor D = 2[1]. What is L(D)? What does it mean? It*
*is nothing but the set of meromorphic functions with at most a pole of*
*order 2 at [1] and no other poles. More precisely,*

*L(D) = {g(z)/(z − 1)*^{2}*|deg(g(z)) ≤ 2}.*

*Because if deg(g(z)) ≥ 3 then it gives a pole at ∞.*

We recall some definitions and properties of divisors.

*Definition 2.5. Let D =* P

*n*_{i}*P*_{i}*be a divisor. We said that D is*
*effective if n*_{i}*≥ 0 for all i, denoted D ≥ 0.*

*WE write D*_{1} *≥ D*_{2} *if D*_{1}*− D*_{2} *≥ 0.*

*Definition 2.6. Given divisors D, D*^{0}*, they are said to be linearly*
*equivalent, denoted D ∼ D*^{0}*, if D − D*^{0}*= div(f ) for some f ∈ M(X).*

*The linear series |D| is thus defined to be the set*

*|D| := {D*^{0}*∈ Div(X)|D*^{0}*≥ 0, D*^{0}*∼ D}.*

Notice that we have a induced natural map
*π : L(D) − {0} → |D|,*

*f 7→ div(f ) + D.*

*If one identify L(D) ∼*= C^{n}*, then |D| can be identified as P** ^{n−1}*. In

*particular, one has that if L(D) 6= 0, then*

*dim |D| = dim L(D) − 1.*

*It is interesting an important to determine the dimension of L(D),*
*denoted l(D). We recall the following fact.*

*{p1}* *Proposition 2.7. Let D =* P

*n*_{i}*P*_{i}*be a divisor. We said that D*
*is effective if n*_{i}*≥ 0 for all i, denoted D ≥ 0. Suppose now that D is*
*an effective non-zero divisor then L(−D) = {0}.*

We leave the proof as an exercise. (Hint: prove that a holomorphic function on a compact Riemann surface must be constant).

*Among all divisors, there is a most important one, called canonical*
*divisor, denoted K** _{X}*. It is a divisor associate to meromorphic 1-form.

*Example 2.8. Let X = C ∪ {∞} be the Riemann sphere. X can be*
*covered by coordinate charts (U*_{0}*, z), (U*_{1}*, w), where U*_{0} *= X − {∞} ∼*= C
*and U*1 *= X − {0} ∼= C. Note that w =* ^{1}_{z}*.*

*We consider a holomorphic 1-form dz on U*0*. This can be extended*
*into a meromorphic 1-form on U*1*, because near ∞, z =* _{w}^{1} *and hence*
*dz =* ^{−1}* _{w}*2

*dw. Therefore, the extended meromorphic 1-form gives rise to*

*a divisor −2[∞]. This is a canonical divisor of X.*

2. DIVISORS 5

Warning. One can pick other meromorphic 1-form and then pro-
duce a different canonical divisor. However, all these are linearly equiv-
alent. (Exercise: check it). Therefore, to be more precise, one should
consider the equivalent classes of divisors of 1-forms. Nevertheless, in
most real application, the quantity that we are going to compute is
the same inside the equivalent class. Therefore, we usually abuse the
*notation by picking any one inside the class and call it the canonical*
*divisor K** _{X}*.

*In general, l(D) can be computed or estimated by the most impor-*
tant theorem for curves, the Riemann-Roch theorem:

Theorem 2.9 (Riemann-Roch).

*l(D) − l(K*_{X}*− D) = deg(D) + 1 − g(X),*

*where K*_{X}*denotes the canonical divisor, l(D) denotes the dimension*
*of L(D), and g(X) is the genus of the curve X.*

We will leave the sketch of the proof to next section. In this section, we concentrate on its applications.

We now redo the example of elliptic curve with the help of Riemann- Roch theorem.

*Let E = C/Λ be an elliptic curve. Then the genus is 1. The 1-from*
*dz on C is doubly periodic, hence induced a 1-form dz on E. Thus*
*K** _{X}* = 0.

*Let D
0 be an effective divisor. By Prop. 2.7, K = 0, and*
Riemann-Roch, we have:

*l(D) = deg(D).*

*Let ¯0 be the image of π(0) in E.The vector space L(k[¯0]) has a nat-*
*ural basis {1}, {1, P}, {1, P, P*^{0}*} respectively when k = 1, 2, 3. How-*
*ever, when k = 6, one finds that l(6[¯0]) = 6 and thus {1, P, P*^{0}*, P*^{2}*, PP*^{0}*,*
*P*^{3}*, P*^{02}*} must be linearly dependent. Hence there must be a relation*
*between them involving P*^{3}*, P** ^{02}*.

*By using L(3[¯0]), one produces a meromorphic map*
*ϕ : E 99K P(L(3[¯0])) = P*^{2}*.*

The linear dependency shows that the image satisfies a cubic polyno- mial. This recover the embedding given in the previous section.

*Turning to a compact Riemann surface X in general. Following the*
*above construction, we would like to ask if there is a divisor D such*
*that L(D) has enough sections in the sense that*

*ϕ*_{D}*: X 99K P(L(D)),*
is an embedding.

By Riemnann-Roch Theorem, one can prove that:

6 1. PRELIMANARY

*Theorem 2.10. Let X be a compact Riemann surface of genus g.*

*Suppose that D is a divisor with deg(D) ≥ 2g + 1, then ϕ**D* *is en*
*embedding.*

Sketch. ¤

3. Riemann-Roch Theorem 4. Algebraic Varieties

CHAPTER 2

### Algebraic Surfaces

1. Riemann-Roch Theorem on Surfaces

*Let D be a divisor on X. If D*^{0}*∼ D, then O(D) ∼= O(D** ^{0}*). Thus

*sometime it’s useful to pick a better element in |D| instead of looking*

*at D itself.*

*Theorem 1.1 (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 D*_{0}*, D*_{1} *∈ |D| with local equation f, f + g on an affine*
*open set U ⊂ X. We have a subseries (a pencil) D*_{λ}*∈ |D| locally*
*defined by f + λg. Suppose that P*_{λ}*∈ U is a singular point of D** _{λ}* not

*in the base locus B := Bs|D|. We may assume that g(P*

_{λ}*) 6= 0. We*have

*f + λg(P*_{λ}*) = 0,*

and *∂*

*∂z*_{i}*(f + λg)(P*_{λ}*) = 0,*
*for all z*_{i}*. Where z*_{1}*, ..., z** _{n}* are the local coordinates.

*These equations defines a subvariety in Z ⊂ U × P*^{1}. And let
*V = pr*_{1}*(Z) ⊂ U.*

*One note that f /g is locally constant (−λ) on V − B. Hence for λ*
*different from value of f /g, D*_{λ}*is non-singular away from B (in U).*

*Next one notice that one can cover X by finitely many affine open*

sets. ¤

*Remark 1.2. If |D| is very ample, then it is base point free.*

*Definition 1.3. A divisor D is said to be ample if mD is very*
*ample for some m > 0.*

Our first aim is the following:

*{decomp}*

*Theorem 1.4. 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 1.5. Let D be a divisor on a projective variety X.*

*Then there are non-singular very ample divisor Y*_{1}*, Y*_{2} *such that D ∼*
*Y*_{1}*− Y*_{2}

7

8 2. ALGEBRAIC SURFACES

*Lemma 1.6. The following are equivalent:*

*(1) D is ample.*

*(2) For every coherent sheaf F on X, we have H*^{i}*(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 gen-*
*erated for all n À 0.*

*{crit1}*

*Remark 1.7. A sheaf is globally generated if the natural map H*^{0}*(X, F)⊗O**X* *→*
*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.*

*Exercise 1.8. Show that if D*_{1} *is very ample and D*_{2} *is base point*
*free, then D*_{1}*+ D*_{2} *is very ample.*

*(Hint: consider the subspace L(D*_{1}*)⊗L(D*_{2}*) ⊂ L(D*_{1}*+ D*_{2}*). Show*
*that the map defined by this subspace is everywhere defined and an*
*embedding. Thus the map defined by D*_{1}*+ D*_{2} *is an embedding.*

*proof of theorem 1.4. X is projective, then X ,→ P** ^{n}* for some

*n. Take H a hyperplane in P*

^{n}*, then H ∩ X is a very ample divisor on*

*X. By abuse the notation, we still called it a hyperplane section and*

*denote it as H.*

Note that very ample is clearly ample. Hence by the Lemma 1.7
*(3), there is an n*_{0}*such that D+n*_{0}*H is base point free. By the exercise,*

*D + (n*_{0} *+ 1)H is very ample.* ¤

We are now able to define intersection of subvarieties. We start by considering intersection on surface.

*Theorem 1.9. 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. (C*_{1}*+ C*_{2}*).D = C*_{1}*.D + C*_{2}*.D,*

*(4) it depends only on the linear equivalence classes. i.e. if C*1 *∼*
*C*2 *then C*1*.D = C*2*.D.*

Proof. See [Ha, V 1.1]. ¤

*Remark 1.10. Let X be a projective variety. An 1-cycle is a formal*
*linear combination of irreducible curves. The group of all 1-cycles is*
*denoted Z*_{1}*(X) (=free abelian group on irreducible curves). One can*
*similarly define a pairing Z*_{1}*(X) × Div(X) → Z.*

*Two curves C*_{1}*, C*_{2} *are said to be numerically equivalent if C*_{1}*.D =*
*C*2*.D for all D, denoted C*1 *≡ C*2*. We define*

*N*_{1}*(X) := Z*_{1}*(X)⊗R/ ≡ .*

*It’s a famous theorem asserts that N*_{1}*(X) is finite dimensional. Its*
*dimensional is called Picard number, denoted ρ(X).*

1. RIEMANN-ROCH THEOREM ON SURFACES 9

*Remark 1.11. Let V ⊂ X be a subvariety of codimension i, and*
*D is a divisor. Then it make sense to consider V.D*^{i}*by decomposing*
*D ∼ H*_{1}*− H*_{2} *and then compute (V ∩ H*_{i}*).D*^{i−1}*in V ∩ H*_{i}*inductively*
*on dimension. One can simply set*

*V.D*^{i}*:= (V ∩ H*1*).D*^{i−1}*− (V ∩ H*2*).D*^{i−1}*.*

*Remark 1.12. Let X be a variety over C. A divisor D gives a*
*class c*1*(D) ∈ H*^{2}*(X, Z) via Div(X) → H*^{1}*(X, O*^{∗}*) → H*^{2}*(X, Z). And*
*a curve C give rise to a class [C] ∈ H*_{2}*(X, Z). The pairing H*_{2}*(X, Z) ×*
*H*^{2}*(X, Z) → Z gives an intersection theory.*

An important feature of ampleness is that it’s indeed a ”numerical property”.

*Theorem 1.13 (Nakai’s criterion). Let X be a projective variety.*

*A divisor D is ample if and only V.D*^{i}*> 0 for all subvariety of codi-*
*mension 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.*

*Another important criterion is due to Kleiman. Let NE(X) ⊂*
*N*_{1}*(X) be the cone generated by effective curves. And let NE(X) be*
its closure.

*For any divisor D, it defines a linear functional on N*_{1}*(X) and we*
*set D*_{>0}*= {x ∈ N*_{1}*(X)|(x.D) > 0}.*

*Theorem 1.14 (Kleiman’s criterion). D is ample if and only if*
*D*_{>0}*⊃ NE(X) − {0}.*

Before we revisit the Riemann-Roch theorem on surface, we need the useful adjunction formula:

*Proposition 1.15 (Adjunction formula). Let S ⊂ X be a non-*
*singular subvariety of codimension 1 in a non-singular variety X. Then*
*K*_{S}*:= K*_{X}*+ S|*_{S}*. In particular, if dimX = 2 then 2g(S) − 2 = (K** _{X}* +

*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

*n*_{i}*D** _{i}*, one

*might want to consider naively that D|*

*:=P*

_{Y}*n*_{i}*(D*_{i}*∩ Y ). But what*
*if D*_{i}*= Y for some i? That is, how to define Y |** _{Y}*?

*One way to think of this is that we deform Y such that lim*_{t→0}*Y** _{t}* =

*Y , then we take Y |*

*:= lim*

_{Y}

_{t→0}*Y*

_{t}*|*

*. (This needs some extra care).*

_{Y}proof of adjunction formula. Recall that a canonical divisor
*is a divisor defined by n-forms if dimX = n. Thus one has Ω*^{n}_{X}*∼*=
*O*_{X}*(K** _{X}*). Where Ω

^{n}

_{X}*denote the sheaf of n-forms on X. Also one can*

10 2. ALGEBRAIC SURFACES

*consider sheaf of n-forms on X with pole along S, denoted Ω*^{n}_{X}*(S). It’s*
clear that Ω^{n}_{X}*(S) ∼= O**X**(K**X* *+ S).*

One has the following exact sequence

*0 → O*_{X}*(K*_{X}*) → O*_{X}*(K*_{X}*+ S) → O*_{S}*(K*_{X}*+ S|*_{S}*) → 0.* *(2.2.1)*
On the other hand, one has the Poincar´e residue map

Ω^{n}_{X}*(S) ³ Ω*^{n−1}_{S}

with kernel Ω^{n}* _{X}*. Hence we have an exact sequence

*0 → Ω*^{n}_{X}*→ Ω*^{n}_{X}*(S) ³ Ω*^{n−1}_{S}*→ 0* *(2.2.2)*
Comparing these two sequences, one sees that Ω^{n−1}_{S}*∼= O**S**(K**X* *+ S|**S*).

*Hence the canonical divisor K**S* *= K**X* *+ 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 z*_{1}*, .., z*_{n}*be the local coordinates of U.*

The sheaf Ω^{n}_{X}*(S) on U can be written as ω =* ^{g(z)dz}_{f (z)}^{1}^{∧...∧d}^{n}*. Since S*
is non-singular, then at least one of _{∂z}^{∂f}

*i* *6= 0. The residue map send ω*
to

*ω*^{0}*:= (−1)*^{i−1}*g(z)dz*_{1}*∧ ... ∧ cdz*_{i}*∧ ... ∧ d*_{n}

*∂f /∂z*_{i}*|*_{f =0}*.*
*This is independent of choice of i since on S*

*df =* *∂f*

*∂z*_{1}*dz*1*+ ... +* *∂f*

*∂z*_{n}*dz**n* *= 0.*

*Another way to put it is that the residue map sends ω to ω** ^{0}* such

*that ω =*

^{df}

_{f}*∧ ω*

*.*

^{0}*It’s clear that the ω*^{0}*= 0 if and only if f (z)|g(z), which means that*
*ω is indeed in Ω*^{n}* _{X}*.

¤ Theorem 1.16 (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, O**X*) + 1

2*D.(D − K**X**).*

*Proof. Write D ∼ H*1*− H*2 *with H**i* are non-singular very ample
divisor. We consider the sequences:

*0 → O(D) ∼= O(H*1*− H*_{2}*) → O(H*_{1}*) → O*_{H}_{2}*(H*_{1}*) → 0,*
*0 → O → O(H*_{1}*) → O*_{H}_{1}*(H*_{1}*) → 0.*

It’s clear that

*χ(X, D) = χ(X, H*1*) − χ(H*2*, O**H*2*(H*1))

*= χ(X, O*_{X}*) + χ(H*_{1}*, O*_{H}_{1}*(H*_{1}*)) − χ(H*_{2}*, O*_{H}_{2}*(H*_{1}*)).*

2. BLOWING-UP AND BLOWING-DOWN 11

By Riemann-Roch on curves and adjunction formula,
*χ(H*1*, O**H*1*(H*1*)) = H*1*.H*1*+ 1 − g(H*1*) = H*1*.H*1*+ 1 −*1

2*(K**X**+ H*1*).H*1*,*
*χ(H*2*, O**H*2*(H*1*)) = H*1*.H*2*+ 1 −* 1

2*(K**X* *+ H*2*).H*2*.*
Collecting terms, one has

*χ(X, D) = χ(X, O**X*)+1

2*(H*1*−H*2*).(H*1*−H*2*−K**X**) = χ(X, O**X*)+1

2*D.(D−K**X**).*

¤ 2. Blowing-up and Blowing-down

*Remark 2.1. The construction of blowing up can be found almost*
*in any book. (Some called it σ-process however). We refer [Beauville,*
*complex algebraic surfaces, chap. II]. However, Beauville only proved*
*that the map h is a bijective morphism. It would be a good exercise to*
*prove that h indeed an isomorphism.*

In this section, we introduce the important notion of blowing-up.

This process is essential in studying singularities and hence birational geometry in general.

We first introduce the local version. Let A* ^{n}* be the affine space

*with coordinates z*

_{0}

*, ..., z*

_{n−1}*and 0 ∈ A*

*be the ”origin”. We construct*

^{n}*a variety Y ⊂ A*

^{n}*× P*

^{n−1}*by {z*

_{i}*X*

_{j}*= z*

_{j}*X*

_{i}*}*

_{i6=j}*, where X*

_{0}

*, ..., X*

*are the homogeneous coordinates of P*

_{n}*. There is a natural morphism*

^{n−1}*π : Y → A*

^{n}*by projection. One sees that π*

^{−1}*(0) ∼*= P

^{n−1}*and π :*

*Y − π*

^{−1}*(0) ∼*= A

^{n}*− {0}. We say Y is the blowing-up of A*

*at 0 and*

^{n}*denoted Bl*0

*(Y ).*

*In general, let x ∈ X be a point in a variety X. Pick an open affine*
*neighborhood U of x. We identify (U, x) with an open set (U*^{0}*, 0) ⊂ A** ^{n}*.
Then one has e

*U := π*

^{−1}*(U*

^{0}*) → U*

^{0}*which is the blowing-up of U*

*at 0.*

^{0}*Glue X − U and eU together, we get π** _{X}* : e

*X → X. Which is called the*

*blowing-up of X at x. Note that one has similarly that π*

^{−1}

_{X}*(x) ∼*= P

^{n−1}*and π*

*: e*

_{X}*X − π*

_{X}

^{−1}*(x) ∼= X − {x}. The divisor π*

_{X}

^{−1}*(x) is called the*

*exceptional divisor, and usually denoted E.*

*Exercise 2.2. Let π : X = Bl** _{x}*(P

^{2}

*) → P*

^{2}

*be the blowing-up of P*

^{2}

*at a point x ∈ P*

^{2}

*. Prove that*

*K*_{X}*= π*^{∗}*K*_{P}^{2} *+ E*
*by local coordinate computation.*

*In fact, if dimX = 2, π : eX → X is a blowing-up at a point x ∈ X,*
then

*K**X*˜ *= π*^{∗}*K*_{X}*+ E.*

12 2. ALGEBRAIC SURFACES

*More generally, if dimX = n, and π : eX = Bl*_{x}*(X) → X is the blowing-*
*up at x, then*

*K**X*˜ *= π*^{∗}*K**X* *+ (n − 1)E.*

Let’s play a little bit around the blowing-ups. Let’s restrict our-
selves to surfaces. One might expect that there are similar higher-
*dimensional formulation. Let X be a surface, and C ⊂ X be a curve.*

*Let f be the local equation of C around x. By fixing local coordinates*
*z*_{1}*, z*_{2}, we can write

*f = f (z*_{1}*, z*_{2}*) = f*_{m}*+ f*_{m+1}*+ ...*

*with f*_{m}*6= 0. We define the multiplicity of C at x to be*
*m*_{x}*(C) := m.*

One can have an equivalent definition by vanishing order of partial
*differentials. Hence one can check the m**x**(C) is well-defined.*

*We consider π : eX = Bl*_{x}*(X) → X. And let C be a curve passing*
*through x ∈ X. Then π*^{−1}*(C) consists of irreducible components, E*
*and the other part maps onto C. The part maps onto C can be defined*

as *C := π*e ^{−1}*(C − {x}),*

*which is called the proper transform of C. Thus we have π*^{−1}*(C) =*
*C ∪ E. More precisely, by computing the equations, one has*e

*π*^{∗}*C = eC + m*_{x}*(C)E,*
*this is called the total transform of C.*

We here collect some properties regarding the blowing-up on sur- face.

*Proposition 2.3. Let π : eX = Bl*_{x}*(X) → X be the blowing-up at*
*x ∈ X. Then one has:*

*(1) There is a natural isomorphism Div(X) ⊕ ZE → Div( eX) by*
*(D, nE) 7→ π*^{∗}*D + nE. And the isomorphism induces an iso-*
*morphism PicX ⊕ ZE → Pic eX.*

*(2) Let D, D*^{0}*∈ Div(X), then (π*^{∗}*D).(π*^{∗}*D*^{0}*) = D.D*^{0}*.*
*(3) Let D ∈ Div(X), then (π*^{∗}*D).E = 0.*

*(4) E.E = −1.*

Proof. It’s easy to check the isomorphism given in (1).

*For (2) and (3), it follows by choosing ∆, ∆** ^{0}* which are linear equiv-

*alent to D, D*

^{0}*respectively but not passing through x.*

*For (4), by adjunction formula and the fact the E ∼*= P^{1},

*−2 = deg(K**E**) = (K**X*˜ *+ E).E = (π*^{∗}*K**X* *+ 2E).E = 2E.E.*

¤ The blowing-up gives the first example of binational morphism.

2. BLOWING-UP AND BLOWING-DOWN 13

*Definition 2.4. By a rational map f : X 99K Y from X to Y , we*
*mean a regular function on a dense Zariski open (or simply non-empty*
*Zariski open) set U ⊂ X.*

*More precisely, a rational map can be written as (U, f ) where U ⊂*
*X is a dense Zariski-open set and f : U → Y is regular.*

*We say (U, f ) ∼ (V, g) if f = g on U ∩ V . In fact, a precise*
*definition of rational map should be the equivalent class of the pairs*
*(U, f ). However, we usually abuse the notation if no confusion is likely.*

*Definition 2.5. A rational map φ : X 99K Y is said to be bira-*
*tional if it admits an inverse. That is, there is an ψ : Y 99K X such*
*that ψ ◦ φ = id*_{X}*, φ ◦ ψ = id*_{Y}

*Example 2.6. Let π : Y = Bl*_{0}(A^{2}*) → A*^{2}*, take ψ : A*^{2}*−{0} → Y ⊂*
A^{2}*×P*^{1} *such that ψ(x, y) = ((x, y), [x, y]). Then ψ◦π = id*_{Y}*, π◦ψ = id*_{X}*.*
*Hence π is a birational morphism.*

*Exercise 2.7. The following are equivalent:*

*(1) X and Y are birationally equivalent.*

*(2) there are non-empty open subset U ⊂ X and V ⊂ Y such that*
*U, V are isomorphic.*

*(3) K(X) ∼= K(Y ) as k-algebra.*

*Given a variety X, one can obtain various birational equivalent*
varieties

*... → X*_{n}*→ X*_{n−1}*→ ... → X*_{1} *→ X*

*by successive blowing-ups. It’s also a natural question to ask if X is*
*obtained by blowing-ups? Another way to put it is if X minimal or*
*not? The precise formulation of minimal model in any dimension is*
quite subtle.

We start by working on contraction on surfaces. In order to produce
*a minimal object, we need to tell whether a surface X is obtained from*
blowing-ups.

*Definition 2.8. Let C ⊂ X be a curve on X, we say that C is a*
*(−1)-curve if C ∼*= P^{1} *and C*^{2} *= −1*

*We seen that we can have a (−1)-curve by blowing-up. In fact we*
*will prove that any (−1)-curve comes from blowing-ups.*

*Theorem 2.9 (Caltelnuovo). Let X be a surface (non-singular*
*complex projective surface) with E ⊂ X a (−1)-curve. Then there*
*is a morphism π : X → X*^{0}*with X*^{0}*non-singular such that π is the*
*blowing-up of X*^{0}*with exceptional divisor E.*

Proof. The idea is to construct a morphism which is identical at
*E but isomorphic outside E.*

*First pick H any very ample divisor on X, Let k := H.E > 0. We*
*consider H*^{0}*= H + kE, then H*^{0}*.E = 0. Notice that the restriction map*

*H*^{0}*(X, O(H*^{0}*)) → H*^{0}*(E, O(H*^{0}*|*_{E}*) = O*_{E}*) ∼= H*^{0}(P^{1}*, O) ∼= C.*

14 2. ALGEBRAIC SURFACES

*Hence the map ϕ**H*^{0}*produce by |H*^{0}*| is constant on E. We need to*
*refine H so that ϕ**H*^{0}*is isomorphic outside E.*

*To this end, we first pick any very ample H*_{0}*. It’s clear that nH*_{0}
*is very ample for all n > 0. On the other hand, H*_{0} is ample, one can
*arrange that H := nH*_{0} *is very ample with H*^{1}*(X, O(H)) = 0.* ^{1}

Consider the exact sequence

*0 → O*_{X}*(H+(i−1)E) → O*_{X}*(H+iE) → O*_{E}*(H+iE|*_{E}*) = O*_{E}*(k−i) → 0.*

*Claim. H*^{1}*(X, O(H + iE)) = 0 for all 1 ≤ i ≤ k.*

Grant this for the time being, then one has an exact sequence

*0 → H*^{0}*(X, O*_{X}*(H+(i−1)E)) → H*^{0}*(X, O*_{X}*(H+iE)) → H*^{0}*(E, O*_{E}*(k−i)) → 0.*

*Note that H*^{0}*(E, O**E**(k−i)) is of dimension k−i+1, let a**i,0**, ..., a**i,k−i* *∈*
*H*^{0}*(X, O(H + iE)) be the lifting of a basis in H*^{0}*(E, O(k − i)).*

Remark. Before we move on, we would like to remark the dif-
*ference between H*^{0}*(X, O(D)) and L(D). It actually comes from two*
*possible definition of O(D). If we define the sheaf O(D) as O(D)(U) =*
*{f ∈ K(X)|div(f ) + D|*_{U}*≥ 0 on U}. Then H*^{0}*(X, O(D)) = L(D).*

*However, another way to look at the sheaf O(D) is to consider it as*
*the sheaf of sections line bundle associate to D. Then under this con-*
*sideration, for s ∈ H*^{0}*(X, O(D)), div(s) gives an effective divisor D*_{s}*linearly equivalent to D. To view it as L(D) is the classical treatment.*

The modern viewpoint tends to think it as section of line bundles. We
*take the convention that H*^{0}*(X, O(D)) represents the global section of*
*line bundle of D from now on.*

Let me describe the correspondence in more detail. Given a divisor
*D, one has a system of local equations (U*_{i}*f** _{i}*). The basic idea behind the
notion of line bundle is instead of looking at functions, we look at local
functions satisfying given patching conditions. The correspondence is
given as

*L(D) → H*^{0}*(X, O(D)),*
*f 7→ (U*_{i}*, f f*_{i}*) = s.*

And the correspondence between their divisor is given by
*div(s) = div(f ) + D,*

*which is an effective divisor D*_{s}*∈ |D|.*

*Turning back to the proof, let s ∈ H*^{0}*(X, O(E)) be a section*
*such that div(s) = E. Then the map H*^{0}*(X, O*_{X}*(H + (i − 1)E)) →*
*H*^{0}*(X, O*_{X}*(H + iE)) is given by multiplying s. Therefore, by working*
*on the sequence inductively, one can have a basis of H*^{0}*(X, O(H +kE)),*
given as

*{s*_{0}*s*^{k}*, ..., s*_{n}*s*^{k}*, a*_{1,0}*s*^{k−1}*, ..., a*_{1,k−1}*s*^{k−1}*, ..., a*_{k−1,0}*s, a*_{k−1,1}*s, a*_{k}*}.*

1*This follows by picking n À 0 such that nH*0*− K**X* is ample. By Kodaira
*vanishing theorem, we have H*^{1}*(Xω⊗O(nJ*0*− K**X*)) = 0.

2. BLOWING-UP AND BLOWING-DOWN 15

*We consider the map ϕ**H*^{0}*: X → P** ^{N}* given by the above basis.

*Note that a**k**∈ H*^{0}*(X, O(H*^{0}*)) whose restriction to E is a non-zero con-*
*stant. Hence one has ϕ is well-defined along E and ϕ(E) = [0, ..., a** _{k}*] =

*[0, ..., 1]. Moreover, for x 6∈ E, s(x) 6= 0, hence*

*[s*0*s*^{k}*(x), ..., s**n**s*^{k}*(x)] = [s*0*(x), ..., s**n**(x)] = ϕ**H**(x).*

*Since H is very ample, ϕ*_{H}*defines an embedding on X and hence*
*on X − E. One sees that the first n + 1 coordinate of ϕ*_{H}* ^{0}* gives an

*embedding on X −E already, so it follows that ϕ*

_{H}*gives an embedding*

^{0}*on X − E.*

*It remains to show that X*^{0}*:= ϕ*_{H}^{0}*(X) is non-singular. Let U ⊂ X*
*be the open subset defined by a*_{k}*6= 0. It’s clear that E ⊂ U. We want*
*to identify U with an open set V ⊂ f*A^{2} *⊂ A*^{2}*×P*^{1}. This can be achieved
by considering

*h : U → A*^{2}*× P*^{1}*,*
*x 7→*¡

(*a*_{k−1,0}*s*

*a*_{k}*(x),a*_{k−1,1}*s*

*a*_{k}*(x)), [a*_{k−1,0}*(x), a*_{k−1,1}*(x)]*¢
*.*

*We might need to shrink U so that a*_{k−1,0}*(x) and a*_{k−1,1}*(x) are not*
*simultaneously vanishing. It’s obvious that h factor through f*A^{2}. Let
*V = h(U) ⊂ f*A^{2}. Moreover, one has the commutative diagram

*U* *−−−→ f** ^{h}* A

^{2}

*ϕ*_{H0}

y ^{π}

y
*ϕ*_{H}^{0}*(U)* *−−−→ A*^{¯}^{h}^{2}*,*
*where ¯h = (*^{a}^{k−1,0}_{a}^{s}

*k* *,*^{a}^{k−1,1}_{a}^{s}

*k* ) is a rational map on P^{N}*defined on ϕ*_{H}^{0}*(U).*

*Another remark is that h clearly maps E ⊂ U onto E ⊂ f*A^{2}. It suffices
*to show that h : U → V is an isomorphism. Because, the induced map*

*¯h is an isomorphism . Therefore, ϕ*_{H}^{0}*(U) is non-singular at ϕ*_{H}^{0}*(E),*
which is the only possible singularity.

*However, to show that h is an isomorphism is not trivial. One can*
first prove that it’s a hemeomorphism, hence in particular, bijective.

*Then one prove the h induces isomorphism on all local rings. (cf. [Ha.*

Ex I.3.2, I.3.3]) ¤

*{neg_int}*

*Lemma 2.10. [KM 3.40] Let Y → X be a resolution of a normal*
*surface with Y projective. Let E*_{i}*be the exceptional divisors. Then*
*binary form (E*_{i}*· E*_{j}*) is negative definite.*

*Proof. Let D =* P

*a*_{i}*E*_{i}*. We would like to prove that D*^{2} *< 0.*

*Suppose on the contrary that D*^{2} *≥ 0. Assume that D is effective.*

*Pick H an ample divisor on Y such that H − K**Y* is ample and
*H*^{2}*(Y, O(H + nD)) = 0 for all n À 0.*

Then by Riemann-Roch, we have

*h*^{0}*(Y, O(H + nD)) − h*^{1}*(Y, O(H + nD)) = χ(Y, O(H + nD)),*

16 2. ALGEBRAIC SURFACES

*goes to infinity as n goes to infinity.*

*However, let f**∗**(nD + H) be the Weil divisor by push-forward. It’s*
clear that

*H*^{0}*(Y, O(nD + H)) ⊂ H*^{0}*(X, O**X**(f**∗**(nD + H))) = H*^{0}*(X, O**X**(f**∗**(H))),*
which is bounded by a finite dimensional space. This is the required
contradiction.

*If D = D*_{+}*+ D*_{−}*, then D*^{2} *≤ (D*^{2}_{+}*+ D*^{2}_{−}*) < 0. We are done.* ¤
3. Minimal Model Program for Surfaces

We will need the following terminology and statements from mini- mal model program.

*Definition 3.1. Let X be a smooth projective variety. It is said*
*to be minimal if K*_{X}*is nef.*

*Suppose that −K**X* is not nef then the Cone Theorem and the
Contraction Theorem in minimal model program asserts the following:

*Theorem 3.2 (Cone Theorem). Let X be a nonsingular projective*
*variety. Suppose that −K*_{X}*is not nef. Then there is a rational curve*
*l in X such that −(dim X + 1) ≤ −K*_{X}*· l < 0.*

*Theorem 3.3 (Contraction Theorem). Let X be a nonsingular pro-*
*jective variety. Suppose that −K**X* *is not nef. Then there is a contrac-*
*tion map ϕ : X → Y which is an algebraic fiber space such that ϕ(l)*
*is a point. We also have that the relative Picard number ρ(X/Y ) = 1.*

*Moreover, a curve C ⊂ X is contracted if and only if C ≡*_{num}*λl for*
*some λ ∈ Q.*

*By applying these two theorem to surfaces with −K** _{X}* is not nef,
we end up with the following three situations.

*I. K**X* *· l = −1.*

*Then −2 = (K**X* *+ l) · l gives that l*^{2} *= −1. In other words, l is a (−1)*
*curve. It’s easy to verify that ϕ contracts only l. Moreover, one can*
*check that ϕ is the blowing down.*

*II. K*_{X}*· l = −2.*

*Then similarly, we have l*^{2} *= 0. We claim that dim Y = 1. Suppose*
*that dim Y = 0, then every curve in X is contracted. In particular, an*
*ample curve h is contracted and h ≡*_{num}*λl. However,*

*0 < h · l = λl*^{2} *= 0.*

*This is a contradiction. Suppose that dim Y = 2. We take a Stein*
*factorization X → Z → Y . We may assume Z is normal and hence*
*X → Z is a resolution. By Lemma 2.10, l is contracted hence l*^{2} *< 0,*
a contradiction.

*We have seen that dim Y = 1 with connected fibers. Let f be a*
*general fiber. One sees that −K*_{X}*· f < 0 and f*^{2} *= 0. It follows that f*
*must be a rational curve. Thus ϕ : X → Y is a ruled surface.*

4. CANONICAL SURFACE SINGULARITIES 17

*III. K**X* *· l = −3.*

*Then we have l*^{2} *= 1. We claim that dim Y = 0. Suppose that dim Y =*
*2, then we argue as above. Suppose that dim Y = 1, then one sees that*
*l*^{2} *= 0, a contradiction. Thus we have dim Y = 0.*

*Now ρ(X) = 1, hence every divisor is proportional to l numerically.*

*In particular, K*_{X}*≡ λl. Adjunction formula gives K*_{X}*≡ −3l. We*
*verify that l is ample. To see this, note that l*^{2} *= 1 > 0. Also, for*
*any curve l 6= C ⊂ X, we have C ≡ αl for some α 6= 0. Since*
*l · C = αl*^{2} *= α ≥ 0, thus α > 0 and l · C > 0.*

*Next we claim that |l| = |K*_{X}*+ 4l| gives an isomorphism to P*^{2}. To
*see this, we verify that h*^{2}*(O*_{X}*) = h*^{0}(Ω^{2}_{X}*) = 0 for K*_{X}*≡ −3l can not*
*be effective. Also h*^{1}*(O*_{X}*) = h*^{0}(Ω^{1}* _{X}*) = 0, otherwise there is a non-

*trivial Albanese map. If the Albanese image has dimension ≥ 2, then*one has a non-zero two-form. If the Albanese image has dimension 1,

*then one has a fibration map with fiber f*

^{2}= 0. Both cases lead to a

*contradiction. We conclude that χ(O) = 1. Thus by vanishing theorem*

*and Riemann-Roch theorem we have h*

^{0}

*(l) = χ(l) = 3.*

*Now by Reider’s theorem, one has |K*_{X}*+ 4l| is base point. In total,*

*|l| produce a morphism*

*ψ : X → P*^{2}*.*

*Again, one can claim that image of ψ has dimension 2. That is, ψ is*
*surjective and generically finite. Let h be the hyperplane section in P*^{2},
we have

*1 = l*^{2} *= ψ*^{∗}*(h)*^{2} *= deg(ψ)h*^{2} *= deg(ψ).*

*Thus ψ is birational. It’s easy to see that there is no exceptional curve*
*since Picard number is one. Hence ψ is an isomorphism. This concludes*
the proof.

4. Canonical Surface Singularities

*Given a non-singular surface of general type, by contracting (−1)*
curves, we might assume that it is non-singular minimal surface. Its
*canonical divisor K*_{X}*is nef and big. That is K*_{X}*· C ≥ 0 for all C and*
*K*_{X}^{2} *> 0.*

In this section, we would like to discuss the canonical model for a surface of general type. In general, it might be singular. We shall discuss he possible singularities as well.

*Theorem 4.1 (Base point freeness theorem). Let X be a non-*
*singular projective variety. D is a nef divisor such that D − K*_{X}*is*
*nef and big. Then |mD| is base point free for m À 0.*

Proof. This is a special case of the well-known Base-Point-Freeness Theorem in Minimal Model Program. We refer the reader to [?] for a

general statement and proof. ¤

18 2. ALGEBRAIC SURFACES

An immediate application to surface of general type is that we have
*a morphism ϕ**m* *: X → P for m À 0. Indeed, one sees that ϕ**m* has
*only connected fiber hence birational by increasing m if necessary (cf.*

[?]). In total, we end up with a birational map
*ϕ : X → Y := ϕ*_{m}*(X) ⊂ P*^{N}*.*

*Y is called the canonical model of X, sometimes denoted X** _{can}*.

*One notices that ϕ(C) = pt if and only if C · K*

*X*= 0.

We would like to discuss this in a more general setting.

*Definition 4.2. A point y ∈ Y is said to be a Du Val singularity if*
*there is a resolution f : X → Y such that K*_{X}*·E*_{i}*= 0 for all exceptional*
*curve E*_{i}*.*

We need to require that the above resolution to be a minimal res-
*olution in the sense there there is no (-1) curve over y.*

We have seen that a canonical model for a surface of general type has at worst DuVal singularities.

*{DuVal}*

*Proposition 4.3. Let f : X → Y 3 y be a resolution of a Du Val*
*singularity y. Let f*^{−1}*(y) = ∪E*_{i}*be the exceptional set. Then E*_{i}^{2} *= −2*
*for all i and* P

*E*_{i}*is a normal crossing divisor.*

*Proof. For an exceptional curve E*_{i}*⊂ X that f (E*_{i}*) = y, by*
*Lemma 2.10, E*_{i}^{2} *< 0. Thus 2p*_{a}*(E*_{i}*) − 2 = (K*_{X}*+ E*_{i}*) · E*_{i}*< 0. It follows*
*that E*_{i}*is a rational curve and E*_{i}^{2} *= −2.*

*Moreover, if E*_{i}*∩ E*_{j}*6= 0 for some i 6= j. We consider C = E*_{i}*+ E** _{j}*.
By Lemma 2.10, we have

*0 > C*^{2} *= E*_{i}^{2} *+ E*_{j}^{2}*+ 2E*_{i}*· E*_{j}*= −4 + 2E*_{i}*· E*_{j}*.*
*Hence E*_{i}*· E** _{j}* = 1.

*Furthermore, suppose P = E**i**∩ E**j* *then P 6∈ E**k* *for any k 6= i, j.*

*To see this, suppose on the contrary that P ∈ E**k*. Then we consider
*C = E**i**+ E**j**+ E**k*. By Lemma 2.10, we have

*0 > C*^{2} *= −6+2E*_{i}*·E*_{j}*+2E*_{i}*·E*_{k}*+2E*_{j}*·E*_{k}*= −4+2E*_{i}*·E*_{k}*+2E*_{j}*·E*_{k}*≥ 0,*

a contradiction. ¤

We will need some easy combinatorics to explore some geometry of surfaces.

Include the file

*Proposition 4.4. Let Γ be a graph and Q be its associate qua-*
*dratic form. Then q is negative definite if and only if Γ is of the type*
*A*_{n}*, D*_{n}*, E*_{6}*, E*_{7}*, E*_{8}*.*

*A rational curve C with C*^{2} *= −2 is called (−2)-curve. Given*
*an exceptional set f*^{−1}*(y) = ∪E** _{i}*, we associate a Dykin diagram Γ as

*following: vertex are E*

_{i}*, edge connecting E*

_{i}*, E*

_{j}*if E*

_{i}*∩ E*

_{j}*6= ∅.*

4. CANONICAL SURFACE SINGULARITIES 19

*The associated quadratic form Q(Γ) is the same as the intersection*
*form on {E**i**}. A singularity which associated Dykin diagram of its*
*minimal resolution is of A − D − E type is called an A-D-E singularity.*

Therefore, Du Val singularities are A-D-E singularities as well.

Next, we would like to give an explicit description of A-D-E singu- larities. We recall some result of Artin’s.

*Theorem 4.5. Let f : X → Y 3 y be a resolution of a surface*
*singularity with exceptional set f*^{−1}*(y) = ∪E*_{i}*.*

*(1) There is a cycle*P

*a*_{i}*E*_{i}*such that a*_{i}*> 0 for all i and Z ·E*_{i}*≤ 0*
*for all i. The minimal one with this property is called the*

*”fundamental cycle” Z.*

*(2) y is rational, i.e. R*^{1}*f*_{∗}*O*_{X}*= 0, if and only if p*_{a}*(Z) = 0.*

*(3) mult(Q) = −Z*^{2}*.*

*(4) the embedding dimension dim m/m*^{2} *= −Z*^{2}*+ 1.*

*Definition 4.6. A surface singularity y ∈ Y is called a rational*
*double point if it is a rational singularities with multiplicity 2.*

It’s easy to write down a fundamental cycle for each A-D-E surface singularities and show that they are rational double points by Artin’s result. Now we would like to show that a rational double point is ana- lytically equivalent to one of the following hypersurface singularities.

*A*_{n}*x*^{2}*+ y*^{2}*+ z** ^{n+1}* = 0

*D*

_{n}*x*

^{2}

*+ y*

^{2}

*z + z*

*= 0*

^{n−1}*E*

_{6}

*x*

^{2}

*+ y*

^{3}

*+ z*

^{4}= 0

*E*

_{7}

*x*

^{2}

*+ y*

^{3}

*+ y*

^{3}

*z = 0*

*E*

_{8}

*x*

^{2}

*+ y*

^{3}

*+ z*

^{5}= 0

*(∗)*

We will need the following four methods repeatedly:

(1) The Weierstrass preparation theorem

*(2) The elimination of the term y*^{n−1}*from the polynomial y*^{n}*+ ....*

*(3) Hensel’s lemma: let f be a formal power series with leading*
*term f**d**. Assume that f**d* *= gh, then f = GH where G, H*
*having leading terms g, h respectively.*

*(4) Let M be a monomial, then uM is equivalent to M by a suit-*
able coordinate change.

*Step 1. By (1) and (2), we have F = (unit) · (x*^{2}*+ f (y, z)).*

*Step 2. If mult*_{0}*f (y, z) ≤ 2, then by (1) and (2), we have f =*
*(unit) · (y*^{2} *+ (unit) · z*^{m}*). This gives equation of A** _{n}* type by (3) and
(4).

*Step 3. If mult*_{0}*f (y, z) ≥ 4. Set X := (p*^{2}*+r*^{−4}*f (qr, r) = 0). Then*
*π : (p, q, r) 7→ (x = pr*^{2}*, y = qr, z = r)*

*maps X → Y is a resolution. Computation shows that it’s not Du Val.*

*We thus assume that mult*_{0}*f (y, z) = 3, we write f = f*_{3} *+ ....*

20 2. ALGEBRAIC SURFACES

*Step 4. Assume that f*3 is not a cube. We may assume that
*f = z(y*^{2}*+ (unit) · z*^{m}*) by (1),(2). This gives D**n* type by (3),(4).

*Step 5. We thus assume that f*_{3} *= y*^{3}*. We write f = y*^{3}*+ yz*^{a}*u** _{a}*+

*z*

^{b}*u*

_{b}*where a ≥ 3, b ≥ 4 and u*

_{a}*, u*

*are either units or zero.*

_{b}*Step 6. Either a ≤ 3 or b ≤ 5.*

*To see this, suppose on the contrary that a ≥ 4 and b ≥ 6, then set*
*X := (p*^{2}*+ q*^{3}*+ qr*^{a−4}*u*_{a}*(pr*^{3}*, qr*^{2}*, r) + r*^{b−6}*(pr*^{3}*, qr*^{2}*, r) = 0. Then*

*π : (p, q, r) 7→ (x = pr*^{3}*, y = qr*^{2}*, z = r)*

*gives a birational morphism. Computation shows that Y is not canon-*
ical.

*Step 7. a ≥ b−1, b = 4, 5. We have f = y*^{3}*+yz*^{a}*u**a**+z*^{b}*u**b**. If a ≥ b,*
*then we make it into f = y*^{3}*+ z*^{b}*u**b**, then we are done. If a = b − 1,*
*then we make it into f = y*^{3}*+ y*^{2}*z*^{b−2}*u + z*^{b}*u** ^{0}* by (2). We can make it

*into f = y*

^{3}

*+ z*

^{b}*u because 2(b − 2) ≥ b. Thus we get E*

_{6}

*, E*

_{8}types.

*Step 8. Finally, we consider f = y*^{3} *+ yz*^{3}*u*_{a}*+ z*^{b}*u** _{b}*. We consider

*a blowup by y = y*

_{1}

*z*

_{1}

*, z = z*

_{1}

*, one sees that f is reducible for it is*

*reducible after blow-up. Hence we assume that f = y(y*

^{2}

*+ yz*

^{2}

*u*

*+*

_{a}*z*^{3}*u** ^{0}*) ¤

Direct computation by these equations shows that there are Du
*Val. Take Y = (x*^{2} *+ y*^{2} *+ z*^{n+1}*= 0) ⊂ A*^{3} *=: A for example. Let*
*π : B → A be the blowing-up with exceptional divisor E and X be the*
*proper transform of Y . One has K*_{B}*= π*^{∗}*K*_{A}*+2E, and X = π*^{∗}*Y −2E.*

*By adjunction, one sees that K**X* *= π*^{∗}*K**Y**. Note that on X, there is*
*a singularity defined by (x*^{2}*+ y*^{2}*+ z*^{n−1}*= 0) if n ≥ 2. Note also that*
*E ∩ X = D*1*+ D*2 *defined by (x*^{2} *+ y*^{2} = 0). Keep blowing-up, we end
up with a resolution ˜*π : ˜X → Y such that K**X*˜ = ˜*π*^{∗}*K*_{Y}*. Hence Y has*
only Du Val singularities. One also verify that its Dykin diagram is
*A** _{n}*.

Therefore we summarize the following:

*Theorem 4.7. The following are equivalent:*

*(1) q ∈ Y is a Du Val singularity.*

*(2) q ∈ Y is an A-D-E singularities.*

*(3) q ∈ Y is a rational double point.*

*(4) q ∈ Y is analytically isomorphic to a hypersurface singularity*
*with equations given above in (∗).*

We have also seen that the singularities appeared in the canonical model of a surface of general type must be Du Val.