##### Document Text Contents

Page 1

ALGEBRAIC GEOMETRY

Page 2

ALGEBRAIC GEOMETRY

Proceedings of the Conference at Berlin

9-15 March 1988

Edited by

H.KDRKE

Sektion Mathematik, Berlin, Germany

and

J. H. M. STEENBRINK

Mathematical Institute, Nijmegen, The Netherlands

Reprinted from

COMPOSITIO MATHEMA TICA

Volume 76, Nos 1 & 2, 1990

KLUWER ACADEMIC PUBLISHERS

DORDRECHT / BOSTON / LONDON

Page 151

150 J. Franke

We have a commutative diagram

P'i'jk'C NT' P'i'kC"C -(-A-) ~) p*(k'j)'q'*C

j"p'*q*k'C--~. i"P'·lq··c

(B) r'k"''j'q .. c NT' (k"i'l'rq··c

i"(qp')*k'C--~) jt!k"'(q'p")*C--~) (k"j')'(q'p")*C

(47)

The commutativity of (A) belongs to the conditions which were used to

characterize the isomorphism j 'k" --+ (k'j)' defined in 4.8, and (B) is of type 3(7)

(applied to the biadmissible functor F = k I). If we insert C = r*(.) in (46) and

apply ii, we get a diagram whose outer contour can be identified with (46).

Now we prove 4.7.3.3. Since (20) is clear for a smooth morphism g, it suffices

to consider the case of a regular closed immersion g. In this case, the proof

consists of two parts:

SUBLEMMA 1. Ifin 0': X ~ Y -4 Zi is a regular immersion and p is smooth, then

i 'p* ( j'(YP.ld) ., , I"P'

1 !"

(pl)~ ) (pi)'

commutes.

SUBLEMMA 2. We suppose that in a Cartesian diagram

i'

X'--~) y'

1 1,

X )y

(48)

Page 152

Chow categories 151

P is smooth and i is a regular immersion. Then IT: X' ..£. y' .4 Y is an admissible

factorization of the lei-morphism ip'. With these notations, the diagram

, base change , F'" ----"'---~) i' 'p*

p' 'i' --~) (ip')' +- (ip')~

(49)

commutes.

It is clear that (20) for a regular immersion g follows from (48) and (49).

Proof of Sub lemma 1. In 4.11, we choose for ko the immersion of X into the

smooth Z-scheme Y, and put S = Y. Then (37) becomes

YxY

/ '\op,

YxY Y

V ~,/~

X ) Y ) Z

P

(Y x Y = YXz Y, and PI = projection to the first factor).

Hence ei,p is

i'p* -+ i'ptp* -+ i'(PPI)* = (pi)~x<T -+ (pi)'

II

(pi)~

(50)

By the definition made at the beginning of 4.10, the isomorphism (pi)~ -+ (pi)~ x <T

in (50) is Cf>;;;'<T,<T,p,' So it remains to prove Cf><TX<T,<T,Pl = Cf><TX<T,<T' By (4.10(ii), the

following diagram commutes:

(pl")lT! lPaX(1XI'1,O"l,Pl (')' ( pi <T x II X <T

1·······, 1 , ...... ~;:,;:.

(pi)~ x <T '-----( ")' ~"'"'P' .,... pi <T x <T X<T

CPa x (1 x (110' X 0'1P23

where P23: Y x Y x Y -+ Y x Y is projection to the last two factors and S12:

Y x Y x Y -+ Y x Y x Y interchanges the first two factors. Since the triangle on

the right side commutes, the right vertical arrow is the identity. Using this and

Page 302

304 Michal Szurek and Jaroslaw A. Wisniewski

ni*(c&") is a line bundle on [pll, say 0(k). The natural morphism n1ni*(c&") ~ c&" is an

evaluation on every fibre and because the sections of 0 EB 0( -1) are constant (in

particular, they do not vanish), we have an exact sequence

with a line bundle Q as a cokernel. Calculating the Chern classes we obtain (3.10).

Finally, the fact that for C2 ~ 1 this sequence splits follows immediately from the

vanishing of first cohomology groups of appropriate bundles on [pll x [pll. This

proves (3.9).

COROLLARY. For c&" as above, c2 (c&") ~ o.

Proof (3.10) gives the exact sequence

o ~ 0(2,.2 - c2 ) ~ C&"(2, 2) ~ 0(1, c2 + 1) ~ 0

and C2 < 0 would contradict the ampleness of C&"(2, 2). This proves (8) and (11) of

the Theorem.

(3.11) PROPOSITION. If c&" is a Fano bundle and Cl (c&") = (-1, -1), c2 (c&") = 2,

then C&"(1, 1) is globally generated and fits in an exact sequence

o ~ 0( -1, -1) ~ 0$3 ~ C&"(1, 1) ~ O.

Proof By (3.9) we have hi(c&"(O, 1)) = hi(C&"(1,0)) = 0, all i, and hi(C&"(1, 1)) = 3 if

i = 0 and 0 otherwise. Restricting c&" to the ruling D 1 gives

o ~ c&"(0, 1) ~ C&"(1, 1) ~ C&"(1, 1)ID1 ~ O.

The induced evaluation morphism HO(C&"(1, 1)) ~ HO(C&"(1, 1)IDJ is then an iso-

morphism. But C&"(1, 1)IDi is globally generated, so is C&"(1, 1). Since hO(C&"(1, 1)) = 3,

computing the Chern classes of the kernel of the evaluation 0 3 ~ C&"(1, 1) gives

(3.11). Conversely, if the inclusion 0 ~ 0(1,1)3 corresponds to a non-vanishing

section of 0(1,1)3, the quotient is a 2-bundle. To complete our discussion of the

case c2 = 2, we must show that [pl(C&") is Fano. Because C&"(1, 1) is globally

generated, it is nef and H + 2~.J'(1.1) is nef, as well. Therefore, to prove that it is

ample, it is sufficient (by the theorem of Moishezon and Nakai) to check that

H + 2~.J'(l.1) has positive intersections with curves in [pl(C&"). However, if H· C = 0,

then C is contained in a fibre and then ~"(l.l)· C > O.

4. Fano bundles over non-minimal Del Pezzo surfaces

Let us recall that any non-minimal Del Pezzo surface Sk is a blow-up of k points

x i(1 ~ i ~ k ~ 8) on the plane, no three on one line and no six of them on a conic.

Page 303

Fano bundles of rank 2 on surfaces 305

The canonical divisor of Sk has the self-intersection number equal to 9 - k. S 1 is

the same as the Hirzebruch surface F l' Let {3: Sk --+ [p2 be the blow-down

morphism, Ci be the exceptional divisors of {3 and H be the inverse image of the

divisor of a line of [p2. Let C be a Fano bundle on Sk' As in the preceeding sections,

we may assume C to be normalized, i.e.,

Since K· Ci = 1, we may apply the same methods as in Section 3 (using Lemma

1.5 from [12]) to conclude easily that Cle, = (!) EB (!) and consequently C = {3*(C')

with a 2-bundle C' on [p2. Moreover, if C1 (C')· H = 0, then C is trivial. Indeed, let

L be the strict transform of a line L c [p3 that passes through one of the points Xi'

Then K sk • L ~ 2 and in virtue of Lemma 1.5 in [12] we have CIL = (!) EB (!),

therefore CIL = (!) EEl (!) and Van de Ven's theorem shows that C' is trivial, so is C.

Let us notice that for k ~ 2 we can always choose a line L that passes through

two of the points Xi' so that -KSk • L = -1 and, as above, c1(C)· L = 0, implying

c1 (C')· H = 0. In other words, we have proved that for k ~ 2, the only ruled Fano

3-fold over a Del Pezzo surface Sk is [pI X Sk' Finally, on the Hirzebruch surface

Fl we have

(a) if c1(C') = 0, then, as above, C = (!) EB (!),

(b) if c1(C') = -1, then, as in (2.3), we infer that C' = (!) EB (!)( -1) or

C' = T,,2( - 2).

References

1. Barth, W., Moduli of Vector Bundles on the Projective Plane. Inv. Math. 42, (1977), 63-91.

2. Demin, I.V., Three-dimensional Fano manifolds representable as line fiberings (Russian). Izv.

Acad. Nauk SSSR, 44, no. 4 (1980). English translation in Math. USSR Izv. 17. Addendum to this

paper in Izv. Acad. Nauk SSSR. 46, no. 3. English translation in Math. USSR Izv. 20.

3. Elencwajg, F. and Forster, 0., Bounding Cohomology Groups of Vector Bundles on P", Math.

Ann. 246 (1980) 251-270.

4. Hartshorne, R., Ample Subvarieties of Algebraic Varieties. Lecture Notes 156 (1970).

5. Hartshorne, R., Stable Vector Bundles of Rank 2 on p 3 • Math. Ann. 238 (1978) 229-280.

6. Kawamata, Y., The cone of curves of algebraic varieties. Ann. Math. 119, 603-633 (1984).

7. Manin, Yu.l Cubic fOfnls, Algebra, Geometry, Arithmetic. North Holland 1974.

8. Mori, Sh., Threefolds Whose Canonical Bundle is not Numerically Effective. Ann. Math. 116,

133-176 (1982).

9. Mori, Sh. and Mukai, Sh.: Classification of Fano 3-folds with B2 ;:. 2. Manuscripta Math. 36,

147-162 (1981).

10. Okonek, Ch., Schneider, M. and Spindler, H.: Vector Bundles on Complex Projective Spaces,

Birkhauser, 1981.

11. Schiffman, B, and Sommese, A. J., Vanishing theorems on complex manifolds. Birkhiiuser 1985.

12. Szurek, M., Wisniewski, J .A., Fano Bundles on p 3 and Q 3. Pacific J ourn. Math. 140, no. 2. (1989).

13. Van de Ven, A., On unifofnl vector bundles. Math. Ann. 195 (1972) 245-248.

ALGEBRAIC GEOMETRY

Page 2

ALGEBRAIC GEOMETRY

Proceedings of the Conference at Berlin

9-15 March 1988

Edited by

H.KDRKE

Sektion Mathematik, Berlin, Germany

and

J. H. M. STEENBRINK

Mathematical Institute, Nijmegen, The Netherlands

Reprinted from

COMPOSITIO MATHEMA TICA

Volume 76, Nos 1 & 2, 1990

KLUWER ACADEMIC PUBLISHERS

DORDRECHT / BOSTON / LONDON

Page 151

150 J. Franke

We have a commutative diagram

P'i'jk'C NT' P'i'kC"C -(-A-) ~) p*(k'j)'q'*C

j"p'*q*k'C--~. i"P'·lq··c

(B) r'k"''j'q .. c NT' (k"i'l'rq··c

i"(qp')*k'C--~) jt!k"'(q'p")*C--~) (k"j')'(q'p")*C

(47)

The commutativity of (A) belongs to the conditions which were used to

characterize the isomorphism j 'k" --+ (k'j)' defined in 4.8, and (B) is of type 3(7)

(applied to the biadmissible functor F = k I). If we insert C = r*(.) in (46) and

apply ii, we get a diagram whose outer contour can be identified with (46).

Now we prove 4.7.3.3. Since (20) is clear for a smooth morphism g, it suffices

to consider the case of a regular closed immersion g. In this case, the proof

consists of two parts:

SUBLEMMA 1. Ifin 0': X ~ Y -4 Zi is a regular immersion and p is smooth, then

i 'p* ( j'(YP.ld) ., , I"P'

1 !"

(pl)~ ) (pi)'

commutes.

SUBLEMMA 2. We suppose that in a Cartesian diagram

i'

X'--~) y'

1 1,

X )y

(48)

Page 152

Chow categories 151

P is smooth and i is a regular immersion. Then IT: X' ..£. y' .4 Y is an admissible

factorization of the lei-morphism ip'. With these notations, the diagram

, base change , F'" ----"'---~) i' 'p*

p' 'i' --~) (ip')' +- (ip')~

(49)

commutes.

It is clear that (20) for a regular immersion g follows from (48) and (49).

Proof of Sub lemma 1. In 4.11, we choose for ko the immersion of X into the

smooth Z-scheme Y, and put S = Y. Then (37) becomes

YxY

/ '\op,

YxY Y

V ~,/~

X ) Y ) Z

P

(Y x Y = YXz Y, and PI = projection to the first factor).

Hence ei,p is

i'p* -+ i'ptp* -+ i'(PPI)* = (pi)~x<T -+ (pi)'

II

(pi)~

(50)

By the definition made at the beginning of 4.10, the isomorphism (pi)~ -+ (pi)~ x <T

in (50) is Cf>;;;'<T,<T,p,' So it remains to prove Cf><TX<T,<T,Pl = Cf><TX<T,<T' By (4.10(ii), the

following diagram commutes:

(pl")lT! lPaX(1XI'1,O"l,Pl (')' ( pi <T x II X <T

1·······, 1 , ...... ~;:,;:.

(pi)~ x <T '-----( ")' ~"'"'P' .,... pi <T x <T X<T

CPa x (1 x (110' X 0'1P23

where P23: Y x Y x Y -+ Y x Y is projection to the last two factors and S12:

Y x Y x Y -+ Y x Y x Y interchanges the first two factors. Since the triangle on

the right side commutes, the right vertical arrow is the identity. Using this and

Page 302

304 Michal Szurek and Jaroslaw A. Wisniewski

ni*(c&") is a line bundle on [pll, say 0(k). The natural morphism n1ni*(c&") ~ c&" is an

evaluation on every fibre and because the sections of 0 EB 0( -1) are constant (in

particular, they do not vanish), we have an exact sequence

with a line bundle Q as a cokernel. Calculating the Chern classes we obtain (3.10).

Finally, the fact that for C2 ~ 1 this sequence splits follows immediately from the

vanishing of first cohomology groups of appropriate bundles on [pll x [pll. This

proves (3.9).

COROLLARY. For c&" as above, c2 (c&") ~ o.

Proof (3.10) gives the exact sequence

o ~ 0(2,.2 - c2 ) ~ C&"(2, 2) ~ 0(1, c2 + 1) ~ 0

and C2 < 0 would contradict the ampleness of C&"(2, 2). This proves (8) and (11) of

the Theorem.

(3.11) PROPOSITION. If c&" is a Fano bundle and Cl (c&") = (-1, -1), c2 (c&") = 2,

then C&"(1, 1) is globally generated and fits in an exact sequence

o ~ 0( -1, -1) ~ 0$3 ~ C&"(1, 1) ~ O.

Proof By (3.9) we have hi(c&"(O, 1)) = hi(C&"(1,0)) = 0, all i, and hi(C&"(1, 1)) = 3 if

i = 0 and 0 otherwise. Restricting c&" to the ruling D 1 gives

o ~ c&"(0, 1) ~ C&"(1, 1) ~ C&"(1, 1)ID1 ~ O.

The induced evaluation morphism HO(C&"(1, 1)) ~ HO(C&"(1, 1)IDJ is then an iso-

morphism. But C&"(1, 1)IDi is globally generated, so is C&"(1, 1). Since hO(C&"(1, 1)) = 3,

computing the Chern classes of the kernel of the evaluation 0 3 ~ C&"(1, 1) gives

(3.11). Conversely, if the inclusion 0 ~ 0(1,1)3 corresponds to a non-vanishing

section of 0(1,1)3, the quotient is a 2-bundle. To complete our discussion of the

case c2 = 2, we must show that [pl(C&") is Fano. Because C&"(1, 1) is globally

generated, it is nef and H + 2~.J'(1.1) is nef, as well. Therefore, to prove that it is

ample, it is sufficient (by the theorem of Moishezon and Nakai) to check that

H + 2~.J'(l.1) has positive intersections with curves in [pl(C&"). However, if H· C = 0,

then C is contained in a fibre and then ~"(l.l)· C > O.

4. Fano bundles over non-minimal Del Pezzo surfaces

Let us recall that any non-minimal Del Pezzo surface Sk is a blow-up of k points

x i(1 ~ i ~ k ~ 8) on the plane, no three on one line and no six of them on a conic.

Page 303

Fano bundles of rank 2 on surfaces 305

The canonical divisor of Sk has the self-intersection number equal to 9 - k. S 1 is

the same as the Hirzebruch surface F l' Let {3: Sk --+ [p2 be the blow-down

morphism, Ci be the exceptional divisors of {3 and H be the inverse image of the

divisor of a line of [p2. Let C be a Fano bundle on Sk' As in the preceeding sections,

we may assume C to be normalized, i.e.,

Since K· Ci = 1, we may apply the same methods as in Section 3 (using Lemma

1.5 from [12]) to conclude easily that Cle, = (!) EB (!) and consequently C = {3*(C')

with a 2-bundle C' on [p2. Moreover, if C1 (C')· H = 0, then C is trivial. Indeed, let

L be the strict transform of a line L c [p3 that passes through one of the points Xi'

Then K sk • L ~ 2 and in virtue of Lemma 1.5 in [12] we have CIL = (!) EB (!),

therefore CIL = (!) EEl (!) and Van de Ven's theorem shows that C' is trivial, so is C.

Let us notice that for k ~ 2 we can always choose a line L that passes through

two of the points Xi' so that -KSk • L = -1 and, as above, c1(C)· L = 0, implying

c1 (C')· H = 0. In other words, we have proved that for k ~ 2, the only ruled Fano

3-fold over a Del Pezzo surface Sk is [pI X Sk' Finally, on the Hirzebruch surface

Fl we have

(a) if c1(C') = 0, then, as above, C = (!) EB (!),

(b) if c1(C') = -1, then, as in (2.3), we infer that C' = (!) EB (!)( -1) or

C' = T,,2( - 2).

References

1. Barth, W., Moduli of Vector Bundles on the Projective Plane. Inv. Math. 42, (1977), 63-91.

2. Demin, I.V., Three-dimensional Fano manifolds representable as line fiberings (Russian). Izv.

Acad. Nauk SSSR, 44, no. 4 (1980). English translation in Math. USSR Izv. 17. Addendum to this

paper in Izv. Acad. Nauk SSSR. 46, no. 3. English translation in Math. USSR Izv. 20.

3. Elencwajg, F. and Forster, 0., Bounding Cohomology Groups of Vector Bundles on P", Math.

Ann. 246 (1980) 251-270.

4. Hartshorne, R., Ample Subvarieties of Algebraic Varieties. Lecture Notes 156 (1970).

5. Hartshorne, R., Stable Vector Bundles of Rank 2 on p 3 • Math. Ann. 238 (1978) 229-280.

6. Kawamata, Y., The cone of curves of algebraic varieties. Ann. Math. 119, 603-633 (1984).

7. Manin, Yu.l Cubic fOfnls, Algebra, Geometry, Arithmetic. North Holland 1974.

8. Mori, Sh., Threefolds Whose Canonical Bundle is not Numerically Effective. Ann. Math. 116,

133-176 (1982).

9. Mori, Sh. and Mukai, Sh.: Classification of Fano 3-folds with B2 ;:. 2. Manuscripta Math. 36,

147-162 (1981).

10. Okonek, Ch., Schneider, M. and Spindler, H.: Vector Bundles on Complex Projective Spaces,

Birkhauser, 1981.

11. Schiffman, B, and Sommese, A. J., Vanishing theorems on complex manifolds. Birkhiiuser 1985.

12. Szurek, M., Wisniewski, J .A., Fano Bundles on p 3 and Q 3. Pacific J ourn. Math. 140, no. 2. (1989).

13. Van de Ven, A., On unifofnl vector bundles. Math. Ann. 195 (1972) 245-248.