A restriction theorem for stable rank two vector bundles on P-3

Contents lists available atScienceDirect

Journal

of

Algebra

www.elsevier.com/locate/jalgebra

A

restriction

theorem

for

stable

rank

two

vector

bundles

on

P

Philippe Elliaa,∗_{, Laurent Gruson}b

a_{DipartimentodiMatematica,35viaMachiavelli,44100Ferrara,Italy} b_{L.M.V.,UniversitédeVersailles-StQuentin,45Av.desEtats-Unis,}

78035 Versailles,France

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received4September2014 Availableonline19May2015 CommunicatedbyStevenDale Cutkosky

MSC:

14J60

Keywords:

Ranktwovectorbundles Projectivespace Restriction Plane

LetE beanormalized,ranktwovectorbundleonP3_.LetH

beageneralplane.If E isstablewithc2 ≥ 4,weshowthat

h0_(E

H(1))≤ 2+ c1.Itfollowsthath0(E(1))≤ 2+ c1.Wealso

showthat if E isproperlysemi-stable andindecomposable,

h0_(E

H(1))= 3.

© 2015ElsevierInc.All rights reserved.

1. Introduction

Weworkoveranalgebraicallyclosedfieldofcharacteristiczero.LetE denoteastable, normalized (−1 ≤ c1(E) ≤ 0) rank two vector bundle on P3. By Barth’s restriction

theorem [1]if H is ageneralplane, then EH is stable (i.e.h0(EH)= 0) except ifE is a null-correlationbundle(c1= 0, c2= 1). Inthis noteweprove:

* Correspondingauthor.

E-mailaddresses:phe@unife.it(Ph. Ellia),Laurent.Gruson@uvsq.fr(L. Gruson).

http://dx.doi.org/10.1016/j.jalgebra.2015.05.009

Theorem 1. Let E be a stable, normalized, rank two vector bundle on P3. Assume c2(E)≥ 4. LetH beageneral plane,then:

(a) h0_(E

H(1))≤ 1 ifc1=−1 and

(b) h0_(E

H(1))≤ 2 ifc1= 0.

In particular itfollowsthat h0_{(E(1))≤ 2+ c} 1.

Theideaoftheproofisasfollows:ifthetheoremisnottruetheneverygeneralplane contains auniqueline,L, such thatEL hassplitting type (r,−r + c1),r≥ c2− 1. We

call such a line a“super-jumping line”. Then we show that these super jumping lines areallcontainedinasameplane, H.TheplaneH isveryunstableforE.Performinga reductionstepwithH,wegetacontradiction.

We observe (Remark 6) that the assumptions (and conclusions) of the theorem are sharp.

For sakeof completeness we show (Proposition 7)that ifE is properly semi-stable, indecomposable, thenh0_(E

H(1))= 3 forH ageneralplane.

2. Proofofthe theorem Weneedsomedefinitions:

Definition 2.LetE beastable,normalizedranktwovectorbundleonP3_{.AplaneH is}

stable if EH isstable; itis semi-stable if h0(EH)= 0 but h0(EH(−1))= 0. A planeis

special if h0(EH(−m))= 0 withm> 1.

A lineisgeneralifthesplittingtypeofELis(0,c1).A lineL isasuper jumping line

(s.j.l.)ifthesplittingtypeofEL is(r,−r + c1),with r≥ c2− 1.

Lemma3.LetE beastable,normalizedranktwovectorbundleonP3.Assumec2(E)≥ 4

and h0_{(EH(1))> 2+ c}

1 ifH isageneral plane. Then:

(i) Every stable plane contains a unique s.j.l., all the other lines are general or, if c1= 0,of type(1,−1).

(ii) Asemi-stableplane containsatmost ones.j.l.

(iii) Thereis atmostonespecial plane.

Proof. (i) IfH isastableplaneeverysectionofEH(1) vanishesincodimension two:

0→ OH→ EH(1)→ IZ,H(2 + c1)→ 0 (∗)

Wehaveh0_(I

Z,H(2+ c1))≥ 2+ c1 bytheassumption.Ifc1=−1,Z hasdegreec2 and

theconicshaveafixedline,LH,andthereisleftapenciloflinestocontaintheresidual scheme of Z with respect to LH. It follows thatthe residual scheme is onepoint and thatlength(Z∩ LH)= c2. Soinboth casesthereis aline,LH,containingasubscheme ofZ oflengthc2.Restricting(∗)toLH wegetELH → OLH(1+ c1− c2).Itfollowsthat

thesplittingtypeofELH is(c2− 1,c1− c2+ 1),henceLH isas.j.l.IfL= LH isanother

lineinH,lets be thelengthofL∩ Z.Restricting(∗)toL we get 0→ OL(s− 1) → EL→ OL(c1− s + 1) → 0

Thissequence splitsexcept maybeifc1 = s= 0 (in thiscasethe splittingtypeis(0,0)

or (1,−1)). If L is as.j.l. then s≥ c2, henceL = LH. This shows thata stable plane containsauniques.j.l.Sinces≤ 1 (resp.s≤ 2)ifc1=−1 (resp.c1= 0),alinedifferent

fromLH isgeneralorhassplittingtype(1,−1). (ii)IfH issemi-stablethenwehave

0→ OH → EH → IT ,H(c1)→ 0 (∗∗)

Heredeg(T )= c2.IfL isalineinH lets denotethelengthofL∩ T .From(∗∗)weget

0→ OL(s)→ EL → OL(c1− s) → 0.Thissequence splits, so thesplitting typeof EL is(s,−s+ c1). IfL is as.j.l. thens≥ c2− 1 andL contains asubschemeof lengthat

leastdeg(T )− 1 ofT .Since c2≥ 4,suchas.j.l.isuniquelydefined.Thisshows thatan

unstableplanecontainsatmostones.j.l.

(iii)Wemayassumeh0(EH(−m− 1))= 0.Wehave

0→ OH→ EH(−m) → IX,H(c1− 2m) → 0 (∗∗∗)

IfL is ageneralline ofH (L∩ X = ∅) then EL hassplitting type(m,−m+ c1), with

m> 1.

Let’sshow thatsuch aspecial plane, ifit exists,is unique.Assume H1, H2 aretwo

special planes. Let H be ageneral stable plane. If Li = H ∩ Hi, then L1, L2 are two

linesofH withsplittingtype(ki,−ki+ c1),ki> 1.By(i)thisisimpossible. 2 Remark4.Therefereepointsoutthatpart(ii)ofLemma 3followsalsofrom Lemma4 of[3].

Wearereadyfortheproofofthetheorem.

Proofof Theorem 1. Let U ⊂ P∗₃ be the dense open subset of stable planes. Wehave amap ϕ: U → G(1,3) definedbyϕ(H)= LH where LH isthe uniques.j.l. contained in H.Soϕ givesarationalmapϕ:P∗₃ −−−> G(1,3).Weclaimthatϕ doesn’textend as amorphism to P∗₃. Indeedinthe contrary casewe wouldhaveasection of the inci-dence variety I ={(H,L) | L ⊂ H} → P∗₃. Since I Proj(Ω_P∗

3(1)) (indeed the fiber

at H of Ω_P∗

an injectivemorphismofvectorbundlesOP∗

3 → TP∗3(k),forsomek.Butthereisnotwist

of T_P∗

3 withanon-vanishingsection. This canbe seenbylookingat c3(TP∗3(k)) orwith

the following argument: thequotient wouldbe aranktwo vectorbundle with H1

∗ = 0, hence,byHorrocks’theorem,adirectsumofline bundleswhichisabsurd.

IfH isasingularpointofthe“true”rationalmapϕ,then,byZariski’sMainTheorem,

H contains infinitely many s.j.l. This implies thatH is the unique special plane (and thatϕ hasasinglesingularpoint).Weclaimthateverys.j.l.iscontainedinH.Indeed letR beas.j.l.notcontainedinH.Letz = R∩H.Thereexistsas.j.l.L⊂ H through z.

TheplaneR,L containstwo s.j.l.henceitisspecial:acontradiction.

Since there are ∞2 s.j.l. we conclude that the general splitting type on the special planeH is(c2− 1,−c2+ c1+ 1).Som= c2− 1 i.e.h0(EH(−c2+ 1))= 0 (andthisisthe

leasttwisthavingasection).Nowweperformareductionstep(see[8,Proposition 9.1]). Ifc1= 0 weget

0→ E→ E → IW,H(−c2+ 1)→ 0

whereEisaranktworeflexivesheafwithChernclassesc₁=−1,c₂= 1,c₃= c2

2−c2+1.

Since E isstable,E isstable too. By[8,Theorem8.2]we getacontradiction. Ifc1=−1,sinceEH∗ = EH(1) weget

0→ E(−1) → E → IR,H(−c2)→ 0

where the Chern classes of E are c₁ = 0, c₂ = 0, c₃ = c2

2. Since E is stable E is

semi-stable.By[8,Theorem8.2]weget, again,acontradiction. 2

Remark5.Theargumenttoshowthatϕ doesn’textendtoamorphismistakenfrom[6]. Anotherwayto provethisistoconsider thesurfaces SLdefinedinthefollowingway:if

L is agenerallineeveryplanethroughL is (semi-)stable, thegeneralonebeing stable. So almost everyplane of the pencil containsa unique s.j.l. taking the closure yields a ruled surfaceSL.Then oneshowsthatSL= SDifL,D aregeneralandthenconcludes bylookingatSL∩ SD (see[5]).

Remark6.Theassumptionc2≥ 4 cannotbe weakenedinTheorem 1. Ifc1=−1 every

stable rankvector bundle, E, with c1 =−1, c2 = 2 issuch that h0(EH(1)) = 2 for a

general planeH (see [9]). IfE(1) is associated to four skewlines, then h0_{(EH(1))= 3}

forH general andci(E)= (0,3).

Ontheotherhandaspecialt’Hooftbundle (E(1) associatedtoc2+ 1 disjointlineson

aquadric)isstablewithc1(E)= 0 and,ifc2≥ 4,satisfiesh0(EH(1))= 2 forH general.

Bytheway,Theorem 1givesbackh0_{(E(1))≤ 2 foraninstanton,aresultfirstproved}

byBoehmerandTrautmann(see[2]andalso[10]).

FinallyletE(1) beassociated tothedisjointunionofc2/2 doublelinesofarithmetic

Concerningproperly semi-stablebundles (c1(E)= 0,h0(E)= 0, h0(E(−1))= 0)we

have:

Proposition7.LetE beaproperlysemi-stableranktwovectorbundleonP3_.AssumeE

isindecomposable.If H isageneralplane then h0_{(EH(1))= 3.}

Proof. We have 0 → O → E → IC → 0, where C is a curve (E doesn’t split) with

ωC(4) OC. Twisting and restricting to a general plane: 0 → OH(1) → EH(1) →

IC∩H,H(1)→ 0.Ifh0(IC∩H,H(1))= 0 it followsfromatheoremofStrano [11,4]thatC isaplanecurve, butthisisimpossible (ωC(4) OC foraplanecurve). 2

Remark8.ToapplyStrano’stheoremweneedch(k)= 0 (see[7]).Thepreviousargument givesaquickproofofTheorem 1incasec1=−1,h0(E(1))= 0.Infactthisremarkhas

beenthestartingpointofthis note.

Remark9. Let C be aplane curve of degreed. A non-zero section of ωC(3) OC(d) yields 0→ O → F(1)→ IC(1)→ 0, whereF is astable ranktwo reflexivesheaf with Chern classes (−1,d,d2). If H is a general plane, h0(FH(1)) = 2 if d > 1 (resp. 3 if

d= 1). Similarly,consideringthe disjoint union of aplane curve and of aline,we get stablereflexivesheafwithc1(F)= 0 andh0(FH(1))= 3.SoTheorem 1doesn’tholdfor

stablereflexivesheaves.Theinterestedreadercantrytoclassifytheexceptions. References

[1]W.Barth,Somepropertiesofstablerank-2vectorbundlesonPn_{,Math.Ann.226(1977)125–150.}

[2]W.Boehmer,G.Trautmann,SpecialinstantonbundlesandPonceletcurves,in:LectureNotesin Math.,vol. 1273,Springer,Berlin,1987,pp. 325–336.

[3]I.Coanda,OnBarth’srestrictiontheorem,J.ReineAngew.Math.428(1992)97–110.

[4]Ph.Ellia,Surleslacunesd’Halphen,in:LectureNotesinMath.,vol. 1389,Springer,1989,pp. 43–65. [5]Ph. Ellia,Somevanishing forthecohomology ofstablerank twovectorbundleson P3_,J.Reine

Angew.Math.451(1981)1–14.

[6]D.Franco,Somevanishingsforthecohomologyofrank2stablereflexivesheaveson_P3_andfortheir

restrictiontoageneralplane,Comm.Algebra23 (6)(1995)2281–2289.

[7]R.Hartshorne,Thegenusofspacecurves,Ann.Univ.FerraraSez.VIISci.Mat.XL(1994)207–223. [8]R.Hartshorne,Stablereflexivesheaves,Math.Ann.254(1980)121–176.

[9]R.Hartshorne,I.Sols,StableranktwovectorbundlesonP3_withc

1=−1,c2= 2,J.ReineAngew.

Math.325(1981)145–152.

[10]T.Nuessler,G.Trautmann,MultipleKoszulstructuresandinstantonbundles,Internat.J.Math. 5 (3)(1994)373–388.

[11]R.Strano, Acharacterizationof completeintersectioncurvesinP3_{,Proc.Amer. Math.Soc. 104}