An ampleness criterion for rank 2 vector bundles on surfaces

13 Pages·2017·0.67 MB·English

by  Arnaud Beauville

An ampleness criterion for rank 2 vector bundles
on surfaces
Arnaud Beauville
Universit´eCˆoted’Azur
Cetraro, September 2018
ArnaudBeauville Anamplenesscriterionforrank2vectorbundlesonsurfaces
Main result
S smooth projective complex surface.
N1pSq “ DivpSq{ „ “ NSpSq{torsion “ Zρ.
num
Proposition
E globally generated rank 2 vector bundle on S.
Assume h0pEq ě 4, and N1pSq “ Z¨c pEq. Then :
1
E is ample or E “ O ‘L.
S
‚ The two cases are distinguished by c pEq ą 0 or “ 0.
2
‚ Recall: E is ample iff OPpEqp1q is ample.
For E globally generated, ϕ : PpEq Ñ PpH0pEqq is a morphism
E
(induced by H0pEqbO (cid:16) E); then : E ample ðñ ϕ finite.
S E
‚ In particular, h0pEq ě 4 necessary. But N1pSq “ Zc pEq strong.
1
DE on P2 globally generated not ample, detpEq “ OP2p2q.
ArnaudBeauville Anamplenesscriterionforrank2vectorbundlesonsurfaces
Application I: Lazarsfeld-Mukai bundles
Recall: C Ă S, L P PicpCq globally generated by V Ă H0pC,Lq.
The Lazarsfeld-Mukai bundle E is defined by the exact sequence
C,V
0 Ñ EC˚,V ÝÑ V bCOS ÝÝevÑ L Ñ 0, or dually:
0 Ñ V˚bCOS ÝÑ EC,V ÝÑ NCbL´1 Ñ 0, with NC :“ OSpCq|C ¨
E has rank dimV, c “ rCs, c “ degpLq.
C,V 1 2
Proposition
Assume: dimV “ 2, H1pS,O q “ 0, N1pSq “ Z¨rCs,
S
L and N bL´1 globally generated and fl O . Then:
C C
E is globally generated and ample.
C,V
ArnaudBeauville Anamplenesscriterionforrank2vectorbundlesonsurfaces
Proof :
0 Ñ V˚bC OS Ñ EC,V Ñ NC bL´1 Ñ 0 gives an exact
sequence on H0, hence a commutative diagram:
0 ´Ñ V˚bO ´´Ñ H0pE qbO ´´Ñ H0pN bL´1qbO ´Ñ 0
› S C,V§ S C § S §
› § § §
› § § §
› evđ evđ đ
0 ´Ñ V˚bO ´´´´Ñ E ´´´´´´´Ñ N bL´1 ´´´Ñ 0
S C,V C
ùñ E globally generated.
c pE q “ rCs, c “ degpLq ą 0, h0pEq ě 4 ùñ E ample.
1 C,V 2
Example : S K3 with PicpSq “ Z¨rCs, |L| “primitive” linear series
– i.e. |L| and |K bL´1| base point free.
C
Question : Does the result extend (say, for K3) for dimpVq ě 3?
ArnaudBeauville Anamplenesscriterionforrank2vectorbundlesonsurfaces
Application II: congruences of lines
Let G :“ Gp2,4q Ă P5. A surface S Ă G gives rise to a 2-dimen-
sional family of lines in P3, called a congruence.
The fundamental locus F of the congruence is the set of points
S
in P3 through which pass 8 lines of the congruence.
Proposition
If N1pSq “ Z¨rO p1qs and degpSq ě 2, F “ ∅.
S S
Proof: E :“restriction to S of the universal quotient bundle on G.
E globally generated, h0pEq ě 4 (otherwise S is a plane),
detpEq “ O p1q, c pEq ą 0 ùñ E ample.
S 2
But PpEq “ tpx,(cid:96)q P P3ˆS|x P (cid:96)u. The projection p : PpEq Ñ P3
satisfies p˚OP3p1q “ OPpEqp1q, hence p finite ðñ FS “ ∅.
ArnaudBeauville Anamplenesscriterionforrank2vectorbundlesonsurfaces
Congruences of lines (continued)
Corollary
S “ GXH XH , H very general hypersurface of degree d ,
d1 d2 di i
with pd ,d q ‰ p1,1q or p1,2q ùñ F “ ∅.
1 2 S
Proof : PicpSq “ Z¨rO p1qs by Noether-Lefschetz.
S
Example.´ Perhaps the simplest nontrivial example of congruence
is the family of bisecants to a twisted cubic T Ă P3. Then
S “ Sym2T “ P2 embedded in P5 as a Veronese surface. The
vector bundle E on P2 is globally generated but not ample since
FS “ T. It has h0pEq “ 4, detpEq “ OP2p2q, c2 “ 3.
ArnaudBeauville Anamplenesscriterionforrank2vectorbundlesonsurfaces
Application III: surfaces with ample cotangent bundle (?)
My original motivation was to find new examples of surfaces with
ample cotangent bundle. These surfaces have very interesting
properties, but few concrete examples are known.
Applying the Proposition to Ω1 gives:
S
Corollary
Assume that Ω1 is globally generated, qpSq ě 4 and
S
N1pSq “ Z¨rK s. Then Ω1 is ample.
S S
Unfortunately I do not know any example of such a surface. The
problem is that the condition N1pSq “ Z¨rK s is very difficult to
S
check. Help appreciated!
ArnaudBeauville Anamplenesscriterionforrank2vectorbundlesonsurfaces
Gieseker’s lemma
The proof uses the method of Bogomolov to prove that the
restriction of a stable bundle to a sufficiently ample curve is stable.
The starting point is the following easy observation:
Gieseker’s lemma
E globally generated vector bundle on X projective irreducible.
E not ample ðñ DC Ă X irreducible and u : E (cid:16) O ¨
C
ðù: E ample ñ E is ample ñ any quotient of E ample.
|C |C
ùñ: DC1 Ă PpEq with ϕEpC1q “ tptu, i.e. OPpEqp1q|C1 “ OC1.
Let p : PpEq Ñ S. ϕ : PpE q ãÑ PpH0pEqq ñ p : C1 Ý„ÑC.
E s
ù section s : C Ý„ÑC1 Ă PpEq; under s, p˚E (cid:16) OPpEqp1q pulls
back to E (cid:16) O .
|C C
ArnaudBeauville Anamplenesscriterionforrank2vectorbundlesonsurfaces
Proof of the Proposition:
E globally generated rank 2 vector bundle on S, h0pEq ě 4,
N1pSq “ Z¨rc pEqs. We assume that E is not ample.
1
Gieseker’s lemma: DC Ă S and u : E (cid:16) O ¨ Define:
C
u
0 Ñ F ÝÑ E ÝÝÑ O Ñ 0 p‹q F rank 2 bundle on S.
C
Strategy : We want to show that the discriminant
∆ :“ 4c pFq´c2pFq is ă 0, so that F is unstable (Bogomolov);
F 2 1
more precisely, F, hence also E, contains a positive line bundle.
This will imply that E˚ has a nonzero section, which splits a trivial
factor out of E.
From p‹q we get cpFq “ cpEq¨cpO q´1 “ cpEq¨p1´rCsq.
C
Put c :“ c pEq: c pFq “ c ´rCs, c pFq “ c ´c ¨rCs´rCs2
i i 1 1 2 2 1
hence ∆ “ ∆ ´2c ¨rCs´rCs2.
F E 1
ArnaudBeauville Anamplenesscriterionforrank2vectorbundlesonsurfaces
Proof (continued)
u
0 Ñ F ÝÑ E ÝÝÑ O Ñ 0. p‹q
C
We have found: c pFq “ c ´rCs, ∆ “ ∆ ´2c ¨rCs´rCs2.
1 1 F E 1
Now N1pSq “ Z¨c ùñ C „ rc for some r ě 1. Then
1 num 1
c pFq “ p1´rqc , ∆ “ ∆ ´pr2`2rqc2 “ 4pc ´c2q´pr2`2r´3qc2,
1 1 F E 1 2 1 1
therefore ∆ ď 4pc ´c2q´pr2´1qc2 ă ´pr2´1qc2 by
F 2 1 1 1
Lemma
G globally generated rank 2 on a surface, h0pGq ě 4 and
H1pdetpGq´1q “ 0 ùñ c2pGq ą c pGq.
1 2
ArnaudBeauville Anamplenesscriterionforrank2vectorbundlesonsurfaces