射影線織曲面のイデアル自由分解につ いて

2010

射影線織曲面のイデアル自由分解につ いて

早稲田大学基幹理工学術院 修士課程

2

5109A003-1

石井 晋平 指導教員名 楫 元

Abstract

ideal sheaf

free resolution

Beilinson theorem

ruled surface

free resolution

0

locally free sheaf

ruled surface

free resolution

Contents

1 Introduction 2

2 Preliminaries 4

2.1 Ruled surface . . . . 4

2.2 Theorems . . . . 4

3 Example 7 3.1 Cohomology

. . . . 7

3.2 Beilinson spectral sequence

. . . . 9

3.3 Beilinson monad

. . . . 11

3.4 Summary and problems . . . . 13

1 Introduction

基礎体は複素数体

C

曲線上の

P

-束を ruled surface

rank 2

locally free sheaf

very ample divisor

sheaf

ideal sheaf

free resolution

Macaulay 2

Theorem 1.1 C

g

b

C

divisor, L

C

line bundle

E

C

rank 2 normalized locally free sheaf

0 → O

→ E → L → 0

k

divisor A = kC

+ bf

ruled surface X ∼ = P(E )

very ample

このとき射影空間に

|A|

X

ideal sheaf I

free resolution

このアルゴリズムでは,次のような手順で

free resolution

1.

2.

locally free sheaf

ruled surface

3.

very ample divisor

4.

ruled surface

ideal sheaf

free resolution

[AT]

free resolution

locally free sheaf

そこで本論文では, [AT]の結果を一般化し, 計算機を使わない手法で

ruled surface

free res- olution

spectral sequence

Eisenbud, Floystad, Schreyer [EFS]

Beilinson monad

Beilinson theorem

coherent sheaf

cohomology

coherent sheaf

exact sequence

Free resolution

(手順 1) Ruled surface

very ample divisor

(手順 2) Twisted ideal sheaves

cohomology

(手順 3) Beilinson theorem

resolution

(手順 4) Differential sheaf

free resolution

3)

resolution

free

4

この手法を

C

E

0

locally free sheaf, very ample divisor

A = C

+ 3f

ruled surface

Main result C

g = 1

E

C

rank 2,

0

normalized locally free sheaf

X = P(E )

very ample divisor A = C

+ 3f

P

scroll

sheaf I

tautological section, f

fibre

Definition 2.4

このとき,上記の手順によって

0 −→ O(−6) −→ 6O(−5)⊕6O(−4) −→ 15O (−4)⊕18O(−3) −→ 20O(−3)⊕3O(−2) −→ I

−→ 0

free resolution

O

一方, [AT]でも同じ設定のもとでアルゴリズムを用いて

ruled surface

free resolution

( Example 3.1 ) .

free resolution

E

3

free resolution

[AT]

Example 3.1

その結果, 目標であった曲線の種数, locally free sheafの次数と構造, very ample divisorを与え れば, minimal free resolutionが一意的に定まることは示せなかった. しかし,曲線の種数, locally

free sheaf

free resolution

ideal sheaf

resolution

minimal free resolution

minimal free resolution

free

resolution

2 Preliminaries

2.1 Ruled surface

ここでは

ruled surface

[H]

Definition 2.1 (Ruled surface) C

g

X

ruled surface

π : X → C

y ∈ C

X

:= π

(y)

fibre

X

∼ = P

π

section

(i.e.

morphism σ : C → X

π ◦ σ = id

Proposition 2.2 π : X → C

ruled surface

C

rank 2 locally free sheaf E

X ∼ = P(E )

P(E )

C

ruled surface

C

rank 2 locally free sheaves E , E

P(E ) ∼ = P(E

)

C

invertible sheaf L

E

∼ = E ⊗ L

Proposition 2.3 π : X → C

ruled surface

E

X ∼ = P(E )

H

(E ) 6= 0

deg L < 0

invertible sheaf L

H

(E ⊗ L ) = 0.

さらにこの場合,ある

section σ

: C → X

C

L (C

) ∼ = O

(1)

Definition 2.4 Locally free sheaf E

normalized

E

Proposition 2.3

E

E

e = − deg E

X

invariant

C

Proposition 2.3

(tautological section

f

fibre

PicX = {aC

+ bf |a ∈ Z, b ∈ PicC}, NumX = {aC

+ bf|a, b ∈ Z}

と表せる.ただし,

bf

X

divisor π

b

Lemma 2.5 K

ruled surface X

canonical divisor

K ≡ −2C

+ (2g − 2 − e)f

Definition 2.6 Scroll

ruled surface

fibres f

1

2.2 Theorems

ここでは

resolution

Beilinson [B]

spectral sequence

Theorem 2.7 P

coherent sheaf F

E

-term

E

= H

(P

, F (p)) ⊗ Ω

(−p)

で与えられる

supectral sequence E

E

-term

( E

= 0 (p + q 6= 0)

E

= F

/F

(0 ≤ p ≤ n)

(F

filtration

F = F

⊇ F

⊇ F

⊇ · · · ⊇ F

⊇ F

= 0

次に, Beilinson spectral sequenceの発展形として

Eisenbud, Floystad and Schreyer [EFS]

Beilinson monad

Theorem 2.8 P

coherent sheaf F

B

= M

j

H

(P

, F (e − j)) ⊗ Ω

(j − e)

となるような

complex

B : · · · −→ B

−→ B

−→ B

−→ · · ·

が存在し,

B

B

exact

B

homology

F

B

exact

complex

monad

Remark B

Theorem 2.7

E

B

= M

j

E

が成り立つ. つまり

B

E

p + q = e

以下,計算に必要な諸定理を述べる.

Theorem 2.9 ([BL]) X

C

invariant e ruled surface, D ≡ aC

+ bf

X

divisor

a ≥ 1

D

very ample

b >

( 1 − a/2 (e = −1 and a ≥ 4) ae + 2 (otherwise)

Proposition 2.10 ([H]) Ruled surface X

divisor D

D.f ≥ 0

i ≥ 0

H

(X, L (D)) ∼ = H

(C, π

L (D))

Lemma 2.11 ([H]) F , F

, F

locally free sheaves

→ F

→ F → F

→ 0

exact

r

S

(F )

filtration S

(F ) = F

⊇ F

⊇ F

⊇ · · · ⊇ F

⊇ F

= 0

F

/F

∼ = S

(F

) ⊗ S

(F

)

∧

F

Theorem 2.12 (Kodaira vanishing) X

n

L

X

ample invertible sheaf, ω

X

canonical sheaf

(1) H

(X, L ⊗ ω) = 0 for i > 0;

(2) H

(X, L

) = 0 for i < n.

Theorem 2.13 (Riemann-Roch) X

surface, D

X

divisor, K

X

canonical divisor, p

X

h

(X, D) − h

(X, D) + h

(X, D) = 1

2 D.(D − K

) + 1 + p

Theorem 2.14 ([But]) X = P(E )

ruled surface

O

(1)

X

very ample divisor

scroll

X

projectively normal

Definition 2.15 (mapping cone) B., C .

complexes, f : B. → C .

complex

B. = · · · −→ B

−−−→ B

−−−→ B

−→ · · ·

↓ f ↓ f ↓ f

C . = · · · −→ C

−−−→ C

−−−→ C

−→ · · ·

cone(f ) = · · · −→ B

⊕ C

−→ B

⊕ C

−→ B

⊕ C

−→ · · · d(b, c) = (−d

(b), d

(c) − f (b))

で定義される

complex

mapping cone

Lemma 2.16 B., C .

complexes, f : B. → C.

complex

n

H

(B.) = H

(C.)

cone(f )

exact

3 Example

この章では

[AT]

g = 1, deg E = 0

ruled surface

scroll

Beilinson monad

2

Example 3.1 ([AT]) C

g = 1

E

rank 2

0

normalized locally free

sheaf

E

1. E = O

⊕ O

i.e. P(E ) = C × P

;

2. E = O

⊕ L , where L 6= O

and deg L = 0;

3. non-trivial extention 0 −→ O

−→ E −→ O

−→ 0.

それぞれに対応する

ruled surface P(E )

X

(i = 1, 2, 3)

X

very ample divisor A = C

+ 3f

scroll surface

P

X

free resolution

0 −→ O

(−6) −→ 6O

(−5) −→ 2O

(−3) ⊕ 9O

(−4) −→ 3O

(−2) ⊕ 4O

(−3) −→ I

−→ 0 0 −→ O

(−6) −→ 6O

(−5) −→ 9O

(−4) −→ 3O

(−2) ⊕ 2O

(−3) −→ I

−→ 0 0 −→ O

(−6) −→ 6O

(−5) −→ 9O

(−4) −→ 2O

(−3) −→ 3O

(−2) −→ I

−→ 0

¤

3.1 Cohomology

この節では

Beilinson theorem

cohomology

(手順 1)

X

very ample divisor A = C

+ 3f

P

scroll

Scroll

A = C

+ bf

very ample

b > 2

A = C

+ 3f

h

(X, A) = h

(C, S

(E ) ⊗ O

(3)) = 6

P

(手順 2) h

(P

, I

(2 + p))

I

(2)

Beilinson theorem (Theorem 2.7, 2.8)

E

= H

(P

, I

(2 + p))⊗Ω

(−p)

cohomology

−5 ≤ p ≤ 0, 0 ≤ q ≤ 5

H

Ω

(−p)

0

m = 2 + p

→ I

(m) → O

(m) → O

(m) → 0

0 −→ H

(I

(m)) −→ H

(O

(m)) −→ H

(O

(m))

−→ H

(I

(m)) −→ 0 −→ H

(O

(m))

−→ H

(I

(m)) −→ 0 −→ H

(O

(m))

−→ H

(I

(m)) −→ 0 −→ 0

−→ H

(I

(m)) −→ 0 −→ 0

−→ H

(I

(m)) −→ 0 −→ 0

m

H

(I

(m)) = H

(I

(m)) = 0.

(i) 0 ≤ m ≤ 2

まず,

H

(X, O

(m))

D = m(C

+ 3f )

O

(m) = L (D)

D.f ≥ 0

j

H

(X

, O

(m)) ∼ = H

(C, π

L (D)) = H

(C, S

(E ) ⊗ O

(3m))

h

(C, S

(E ) ⊗ O

(3m))

Lemma 3.2 h

(C, S

(E ) ⊗ O

(3m))

E

(縦軸に 0 ≤ j ≤ 2,

0 ≤ m ≤ 2)

0 0 0

1 0 0

1 6 18

Proof. dim(C) = 1

m

h

(C, S

(E ) ⊗ O

(3m)) = 0

j = 0, 1

(i-i) m = 0

S

(E ) = O

h

(C, S

(E )) = h

(C, O

) = 1, h

(C, S

(E )) = h

(C, O

) = g(C) = 1

(i-ii) m = 1

S

(E ) = E

0 → O

(3) → E ⊗ O

(3) → O

(3) → 0

h

(C, S

(E ) ⊗ O

(3)) = 6, h

(C, S

(E ) ⊗ O

(3)) = 0

(i-iii) m = 2

Locally free sheaf

symmetric product

cohomology

Lemma 2.11

→ O

→ E → O

→ 0

Lemma 2.11

filtration

⊗O(3)

twist

locally free sheaf F

2

exact sequences

0 −→ O

(3) −→ F ⊗ O

(3) −→ O

(3) −→ 0 0 −→ F ⊗ O

(3) −→ E ⊗ O

(3) −→ O

(3) −→ 0

が得られる.これより

h

(C, S

(E ) ⊗ O

(3)) = 18, h

(C, S

(E ) ⊗ O

(3)) = 0

¤ Remark E = O

⊕ O

E = O

⊕ L

exact sequence

0 → O

→ E → O

→ 0

0 → O

→ E → L → 0

Lemma 2.11

最後に, Lemma 3.2から

h

(P

, I

(m))

Lemma 3.3 h

(P

, I

(m))

i = 1, 2, 3

0 ≤ j ≤ 5,

0 ≤ m ≤ 2)

0 0 0

0 0 0

0 0 0

1 0 0

0 0 0

0 0 3

Proof. Cohomology

long exact sequence

j = 2, 3, 4, 5

j = 0, 1

m

h

(I

(m)) − h

(O

(m)) + h

(O

(m)) − h

(I

(m)) = 0

m = 0

H

(O

) ∼ = H

(O

) ∼ = C

h

(I

(m)) = h

(I

(m)) = 0

m ≥ 1

X

projectively normal

h

(I

(m)) = 0

Lemma 3.2

h

(O

(m))

h

(O

(m))

h

(I

(m))

¤

(ii)−3 ≤ m ≤ −1

Lemma 3.4 h

(P

, I

(m))

i = 1, 2, 3

0 ≤ j ≤ 5,

−3 ≤ m ≤ −1)

0 0 0

0 0 0

18 6 0

0 0 0

0 0 0

0 0 0

Proof. m

H

(X

, O

(m))

X

= 2

j ≥ 3

H

(X

, O

(m)) = 0.

Kodaira vanishing (Theorem 2.12)

j < 2

H

(X

, O

(m)) = 0. Riemann-Roch (Theorem 2.13)

h

(X

, O

(m)) = 1

2 D(D − K

) + 1 + p

= 3m(m + 1) (ここで D = m(C

+ 3f ), K

≡ −2C

+ (2g − 2 − e) ≡ −2C

, p

= −g = −1)

H

(P

, O

(m)) = 0

cohomology

long exact sequence

h

(P

, I

(m))

¤

3.2 Beilinson spectral sequence

前節で求めた

cohomology

ideal sheaf

resolution

spectral sequence

theorem

[K]

O

O, Ω

Ω

(手順 3) Beilinson sepectral sequence (Theorem 2.7)

differential sheaf

resolution

H

(I

(−3)) ∼ = C

E

= H

(I

(−3)) ⊗ Ω

(5) ∼ = 18Ω

(5)

E

= H

(I

(2 + p)) ⊗ Ω

(−p)

E

-term

spectral sequence

(縦軸に 0 ≤ q ≤ 5,

−5 ≤ p ≤ 0)

0 0 0 0 0 0

0 0 0 0 0 0

18Ω

(5) 6Ω

(4) 0 0 0 0

0 0 0 Ω

(2) 0 0

0 0 0 0 0 0

0 0 0 0 0 3O

以下,

q = 4, 5

α : 18Ω

(5) → 6Ω

(4)

E

-term

Ker α Coker α 0 0 0 0

0 0 0 Ω

(2) 0 0

0 0 0 0 0 0

0 0 0 0 0 3O

β : Coker α → Ω

(2)

E

-term

Ker α Ker β 0 0 0 0

0 0 0 Coker β 0 0

0 0 0 0 0 0

0 0 0 0 0 3O

E

-term

E

-term

γ : Ker β → 3O

E

-term

Ker α Ker γ 0 0 0 0

0 0 0 Coker β 0 0

0 0 0 0 0 0

0 0 0 0 0 Coker γ

これ以降変化はなしなので,

E

= E

Beilinson spectral sequence (Theorem 2.7)

filtration I

(2) = F

⊇ F

⊇ F

⊇

· · · ⊇ F

⊇ F

= 0

Ker α = Ker γ = 0

F

/F

= 0, F

/F

= 0, F

/F

= 0, F

/F

= Coker β, F

/F

= 0, F

/F

= Coker γ

4

exact sequences

0 −→ 18Ω

(5) −−−→

6Ω

(4) −→ Coker α −→ 0 (1) 0 −→ Ker β −→ Coker α −−−→

Ω

(2) −→ Coker β −→ 0 (2)

0 −→ Ker β −−−→

3O −→ Coker γ −→ 0 (3)

0 −→ Coker γ −→ I

(2) −→ Coker β −→ 0 (4)

exact sequence (4)

0 −→ Ker β −→ 3O −→ I

(2) −→ Coker β −→ 0 (5)

complex 0 → Coker α → Ω

(2) → 0, exact sequence (5)

complex 0 → 3O → I

(2) → 0

homology

0 −→ Coker α −→ Ω

(2) −→ 0

99K

· · · ⊕ · · · 99K

0 −→ 3O −→ I

(2) −→ 0

complex

Coker α → 3O

Ω

(2) → I

(2)

Lemma 2.16

0 −→ Coker α −→ Ω

(2) ⊕ 3O −→ I

(2) −→ 0 (6)

という

exact sequence

exact sequence (1)

0 −→ 18Ω

(5) −→ 6Ω

(4) −→ Ω

(2) ⊕ 3O −→ I

(2) −→ 0 (7)

3.3 Beilinson monad

Ideal sheaf

resolution

(手順 3’) Beilinson monad (Theorem 2.8)

differential sheaf

resolution

E

= H

(I

(2 + p)) ⊗ Ω

(−p)

E

-term

spectral sequence

0 ≤ q ≤ 5,

−5 ≤ p ≤ 0)

0 0 0 0 0 0

0 0 0 0 0 0

18Ω

(5) 6Ω

(4) 0 0 0 0

0 0 0 Ω

(2) 0 0

0 0 0 0 0 0

0 0 0 0 0 3O

ここで

B

= M

j

H

(P

, F (−j)) ⊗Ω

(j) = E

⊕ E

⊕ E

⊕ E

⊕ E

⊕ E

= 3O ⊕Ω

(2)

同様に

B

= 6Ω

(4), B

= 18Ω

(5)

Beilinson monad

0 −→ 18Ω

(5) −→ 6Ω

(4) −→ 3O ⊕ Ω

(2) −→ 0

が

monad (B

= 3O ⊕ Ω

(2)

exact)

B

homology

I

(2)

0 −→ 18Ω

(5) −→ 6Ω

(4) −→ 3O ⊕ Ω

(2) −→ I

(2) −→ 0 (8)

exact sequence

exact sequence (7)

sequence

(手順 4) differential sheaf

free resolution

(手順 3’)

resolution

free

defferential sheaf

free resolution

P

Euler sequence 0 → Ω → 6O (−1) → O → 0

Lemma 2.11

≤ p ≤ 5

0 −→ Ω

−→

µ 6 p

¶

O (−p) −→ Ω

−→ 0

が成り立つ. これをつなげると,求めたい

differential sheaf

free resolution

0 −→ O(−6) −→ 6O(−5) −→ Ω

−→ 0

0 −→ O(−6) −→ 6O (−5) −→ 15O(−4) −→ 20O(−3) −→ Ω

−→ 0

Remark P

Ω

∼ = O(−6)

次に

resolution (8)

short exact sequence

α : 18Ω

(5) → 6Ω

(4)

0 −→ 18Ω

(5) −−−→

6Ω

(4) −→ Coker α −→ 0 (9) 0 −→ Coker α −→ Ω

(2) ⊕ 3O −→ I

(2) −→ 0 (10)

defferential sheaf

free resolution

0 0

↓ ↓

0 −→ 18O(−1) −→ 18Ω

(5) −→ 0

↓ ↓

↓

0 −→ 6O(−2) −→ 36O(−1) −→ 6Ω

(4) −→ 0

↓ ↓ ↓

0 −→ 6O(−2) −→ 18O(−1) −→ Coker α −→ 0

↓ ↓ ↓

0 0 0

ここで,縦の写像

ρ : 18O (−1) → 36O(−1)

18O (−1)

projective module

sheaf

(−1) → 18Ω

(5)

ρ

ρ

cokernel

→ 6O(−2) → 18O (−1) → Coker α → 0

exact

α

free resolution

これと同様に, exact sequence (10)に対して

defferential sheaf

free resolution

0

↓

0 → 6O(−2) → 18O(−1) → Coker α → 0

↓ ↓

↓

↓

0 → O(−4) → 6O(−3) → 15O(−2) −→

20O(−1) ⊕ 3O −→

Ω

(2) ⊕ 3O → 0

&

↓

I

(2)

↓ 0

ここで縦の写像

f : 18O(−1) −→ 20O(−1) ⊕ 3O , g : 6O (−2) → 15O(−2)

I

free resolution

2

complexes

cone (Definition 2.15)

0 −→ 6O(−2) −→ 18O(−1) −→ 0 (11)

0 −→ O(−4) −→ 6O(−3) −→ 15O(−2) −→ 20O(−1) ⊕ 3O −→ I

(2) −→ 0 (12)

Complex (11)

18O(−1)

exact

homology

Coker α

plex (12)

Lemma 3.5 Complex (12)

20O (−1) ⊕ 3O

exact

⊕ 3O

homology

Coker α

Proof. 20O (−1) ⊕ 3O → I

(2)

⊕ 3O

exact

⊕ 3O

homology

d

: 15O(−2) −→ 20O(−1) ⊕ 3O d

: 20O(−1) ⊕ 3O −→ Ω

(2) ⊕ 3O ϕ : 20O (−1) ⊕ 3O −→ I

(2) ψ : Ω

(2) ⊕ 3O −→ I

(2)

⊕ 3O

homology

Ker ϕ/Im d

= Ker ϕ/Ker d

∼ = Ker ψ = Coker α

ϕ/Ker f ∼ = Ker ψ

module

0 0 0

↓ ↓ ↓

0 −→ Ker d

−→ Ker ϕ −→ Ker ψ −→ 0

↓ ↓ ↓

0 −→ Ker d

−→ M −−→

M

−→ 0

↓ ↓

↓

0 −→ N −→ N −→ 0

↓ ↓

0 0

ただし,

M , M

, N

module

d

: M → M

, ϕ : M → N, ψ : M

→ N

¤

Lemma 3.5

complex (11)

(12)

homology

Lemma 2.16

0 → 6O(−2) → 18O (−1) → 0

↓ · · · ⊕ · · · ↓

· · · ⊕ · · · ↓ ↓

0 → O(−4) → 6O(−3) → 15O(−2) → 20O (−1) ⊕ 3O → I

(2) → 0

0 −→ O(−6) −→ 6O(−5)⊕6O(−4) −→ 15O (−4)⊕18O(−3) −→ 20O(−3)⊕3O(−2) −→ I

−→ 0 (13)

exact sequence

X

(i = 1, 2, 3)

ideal sheaf

free resolution

3.4 Summary and problems

前節で求まった

resolution (13)

i = 1, 2, 3

Example

3.1

i

ideal sheaf

free resolution

いる.

0 −→ O(−6) −→ 6O(−5) −→ 2O(−3) ⊕ 9O(−4) −→ 3O(−2) ⊕ 4O(−3) −→ I

−→ 0 0 −→ O (−6) −→ 6O(−5) −→ 9O(−4) −→ 3O(−2) ⊕ 2O (−3) −→ I

−→ 0 0 −→ O(−6) −→ 6O(−5) −→ 9O(−4) −→ 2O(−3) −→ 3O (−2) −→ I

−→ 0

Resolution (13)

f : 18O(−1) → 20O(−1)⊕

3O, g : 6O(−2) → 15O(−2)

0

↓

0 → 6O (−2) → 18O(−1) → Coker α → 0

↓ ↓

↓

↓

0 → O(−4) → 6O (−3) → 15O(−2) → 20O(−1) ⊕ 3O → Ω

(2) ⊕ 3O → 0

& ↓ I

(2)

↓ 0

これらの写像は, [EFS]で得られた

resolution (8)

6Ω

(4) → Ω

(2) ⊕ 3O

twisted ideal sheaf

cohomology

vector space

H

(I

(−2)) ⊗ Ω

(4) −→ (H

(I

) ⊗ Ω

(2)) ⊕ (H

(I

(2)) ⊗ O)

である.この写像は各

X

(i = 1, 2, 3)

i

f , g

i

free resolution

しかし,この写像の具体的な対応を記述することができなかった. [EFS]で得られた

differential sheaf

resolution

([EFS], [DE])

differential sheaf

当初の目標は曲線の種数, locally free sheafの次数と構造, very ample divisorを与えたとき,

minimal free resolution

(各 X

resolution

一方,

g = 1, deg E = 0, very ample divisor

A = C

+ 3f

ideal sheaf

resolution

free resolution (13)

[AT]

Example 3.1

I

free resolution

I

, I

resolution

I

resolution

resolution (13)

による

I

free resolution

Acknowledgements

修士論文を執筆するにあたり, 毎週の熱心にセミナーを見ていただき,数々のご助言をして頂い た,指導教員の楫 元先生に心より感謝を申し上げます. また,学部

3

ご一緒にセミナーをさせていただいた楫研究室の渡辺 究氏,古川 勝久氏,石川 大蔵氏からは日々 の研究でも多くのご助言を頂きました. 他大学でご活躍中の権業 善範氏,ご卒業された眞鍋 岳氏 からも大きな影響をうけました. ここに感謝の意を表します.

References

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[B] A. Beilinson, Coherent sheaves on P

and problems of linear algebra, Funct. Anal. Appl. 12 (1978), 214-216.

[But] D. C. Butler, Normal generation of vector bundles over a curve, J. Diff. Geom. 39(1) (1994), 1-34.

[BL] A. Biancofiore, E. L. Livorni. On the genus of a hyperplane section of a geometrically ruled surface, Ann. Mat. Pur. Appl. Ann. Mat. Pa. 147 (1987), 173-185.

[DE] W. Decker, D. Eisenbud, Sheaf algorithm using the exterior algebra, in Computations in Algebraic Geometry with Macaulay 2, Algorithms and Computations in Mathematics 8, Springer, 2001.

[DES] W. Decker, L. Ein, F.-O. Schreyer, Construction of surfaces in P

, J. of Algebraic Geom.

2 (1993), 185-237.

[E] D. Eisenbud, Commutative algebra with a view toward algebraic geometry, Grad. Texts Math. 150, Springer, 1995.

[EFS] D. Eisenbud, G. Floystad, F.-O. Schreyer, Sheaf cohomology and free resolutions over exterior algebras, Trans. Am. Math. Soc. (2003), 4397-4426.

[H] R. Hartshorne, Algebraic Geometry, Grad. Texts Math. 52, Springer, 1971.

[K]

[OSS] C. Okonek, M. Schneider, H. Spindler, Vector Bundles on Complex Projective Spaces, Prog. Math. 3, Brikhauser, 1980.

[W] C. A. Weibel, An introduction to homological algebra, Cambridge studies in adv. math. 38,

Cambridge univ. press, 1994.