A note on the very ampleness of complete linear systems on blowings-up of P^3

a r X i v :m a t h /0409526v1[mat h.AG ]27Sep24A NOTE ON THE VERY AMPLENESS OF COMPLETE LINEAR SYSTEMS ON BLOWINGS-UP OF P 3CINDY DE VOLDER AND ANTONIO LAF ACE Abstract.In this note we consider the blowing-up X of P 3along r general points of the anticanonical divisor of a smooth quadric in P 3.Given a complete linear system L =|dH −m 1E 1−···−m r E r |on X ,with H the pull-back of a plane in P 3and E i the exceptional divisor corresponding to P i ,we give necessary and sufficient conditions for the very ampleness (resp.base point freeness and non-speciality)of L .As a corollary we obtain a sufficient condition for the very ampleness of such a complete linear system on the blowing-up of P 3along r general points. 1.Introduction In this note we work over an algebraically closed field of characteristic 0.Let P 1,...,P r be general points of the anticanonical divisor of a smooth quadric in P 3and choose some integers m 1≥...≥m r ≥0.Consider the linear system L ′of surfaces of degree d in P 3having multiplicities at least m i at P i ,for all i =1,...,r .Let X denote the blowing-up of P 3alongs P 1,...,P r ,and let L denote the complete linear system on X corresponding to L ′.and the system is called special if dim(L )>max {−1,vdim (L )}.Let Z be a zero-dimensional subscheme of length 2of X ,then L separates Z if there exists a divisor D ∈L such that Z ∩D =∅but Z ⊂D .The system L on X is called very ample if it separates all such Z .The very ampleness of line bundles of blowings-up of varieties has been studied by several authors,e.g.St´e phane Chauvin and the first author [2],Marc Coppens[3,4],Brian Harbourne [8],Mauro C.Beltrametti and Andrew J.Sommese [1].In theorem 5.1(resp.theorem 4.1)we prove that such a system L is very ample (resp.base point free)on X if and only if m r >0,d ≥m 1+m 2+1and 4d ≥m 1+···+m r +3(resp.d ≥m 1+m 2and 4d ≥m 1+···+m r +2).A fundamental tool for proving these results is theorem 3.1which states that a system L with 2d ≥m 1+···+m 4is non-special if d ≥m 1+m 2−1and 4d ≥m 1+···+m r .If r ≤8,the points P i are in general position on P 3and the dimension and base locus of L on X can be determined using the results from [5,6].The techniques used in this note are a generalization of the ones in [5,6]and make use of the results about complete linear systems on rational surfaces with irreducible anticanonical divisor (see [7,8]).12CINDY DE VOLDER AND ANTONIO LAFACE2.Preliminaries and notationLet L3(d)denote the complete linear system of surfaces of degree d in P3.Consider a general quadric¯Q∈L3(2)in P3and let K¯Q denote the canonical class on¯Q. Then we know that−K¯Q is just the linear system on¯Q induced by L3(2),so we can consider D¯Q∈−K¯Q which is smooth and irreducible.Let P1,...,P r be general points of D¯Q and choose integers m1≥···≥m r≥0. By X r we denote the blowing-up of P3along the points P1,...,P r,E0denotes the pullback of a plane in P3,by E i(i=1,...,r)we mean the exceptional divisor on X r corresponding to P i andπ:X r→P3denotes the projection map.On P3,we let L3(d;m1,...,m r)denote the linear system of surfaces of degree d with multiplicities at least m i at P i for all i=1,...,r as well as the corresponding sheaf.By abuse of notation,on X r,L3(d;m1,...,m r)also denotes the invertible sheaf π∗(O P3(d))⊗O X r(−m1E1−···−m r E r)and the corresponding complete linear system|dE0−m1E1−···m r E r|.Analoguously,on P3,L3(d;m n11,...,m n t t)denotes the linear system of surfaces of degree d with multiplicities at least m i at n i of the points on D Q as well as the corresponding sheaf.Again,the same notation is used to denote the associated complete linear system and invertible sheaf on X r.The virtual dimension of the linear system L=L3(d;m1,...,m r)on P3as well as on X r is defined asvdim(L):= d+33 −r i=1 m i+23 −1.The expected dimension of L is then given byedim(L):=max{−1,vdim(L)}.It is then clear that dim(L)≥edim(L)≥vdim(L),and the system L is called special if dim(L)>edim(L).The system L is associated to the sections of the sheaf O P n(d)⊗I Z,where Z= m i p i is the zero-dimensional scheme of fat points. From the cohomology exact sequence associated to0O P n(d)0,we obtain that h i(L)=0for i=2,3.Therefore v(L)=h0(L)−h1(L)−1,so that a non-empty system is special if and only if h1(L)>0.Note that the strict transform Q r of¯Q on X r is a divisor of L3(2;1r),and Q r is just the blowing-up of¯Q along P1,...,P r.So Pic(Q r)= f1,f2,e1,...,e r ,with f1 and f2the pullbacks of the two rulings on¯Q and e1,...,e r the exceptional curves. By L Q r(a,b;m1,...,m r)we denote the complete linear system|af1+bf2−m1e1−...−m r e r|,and,as before,if some of the multiplicities are the same,we also use the notation L Q r(a,b;m n11,...,m n t t).Let B s be the blowing-up of P2along s general points of a smooth irreducible cubic, then Pic B s= h,e′1,...,e′s ,with h the pullback of a line and e′l the exceptional curves.By L2(d;m1,...,m s)we denote the complete linear system|dh−m1e′1−...−m s e′s|.And again,as before,if some of the multiplicities are the same,weVERY AMPLENESS ON BLOWINGS-UP OF P33 also use the notation L2(d;m n t1,...,m n t r).Note that−K B s=L2(3;1s)contains a smooth irreducible divisor which we will denote by D B s.On B s,a system L2(d;m1,...,m s)is said to be in standard form if d≥m1+m2+m3 and m1≥m2≥···≥m s≥0;and it is called standard it there exists a base ˜h,˜e1,...,˜e s of Pic B s such that L2(d;m1,...,m s)=|˜d˜h−˜m1˜e1−...−˜m s˜e s|is in standard form.As explained in[5,§6],the blowing-up Q of the quadric along1general point can also be seen as a blowing-up of the projective plane along2general points,andL Q(a,b;m)=L2(a+b−m;a−m,b−m).So,in particular−K Q=L Q(2,2;1)=L2(3;12)=−K B.Obviously,this means2that our blowing-up Q r can also be seen as a B r+1andL Q r(a,b;m1,m2,...,m r)=L2(a+b−m1;a−m1,b−m1,m2,...,m r). This implies in particular that we can apply the results from[7]and[8].3.Non-specialityTheorem3.1.Consider L=L3(d;m1,...,m r)on X r with2d≥m1+m2+m3+ m4and m1≥m2≥···≥m r≥0.Then h1(L)=0if(1)d≥m1+m2−1and(2)4d≥m1+···m r if r≥9.Proof.We will assume that m r>0,since otherwise we can work on X r′with r′:=max{i:m i>0}.If r≤8then the points P1,...,P r are general points of P3and the statement follows from[5,Theorem5.3].So we assume that r≥9and consider the following exact sequence0L0Then L⊗O Q r=L Q r(d,d;m1,...,m r)=L2(2d−m1;(d−m1)2,m2,...,m r). Proceeding as in the proof of[5,Lemma5.2]it is easily seen that this is a standard class.Moreover L2(2d−m1;(d−m1)2,m2,...,m r).K B r+1=−4d+m1+···+m r≤0,so we can apply[7,Theorem1.1and Proposition1.2]to obtain h1(L⊗O Q r)=0. On the other hand,one can easily check that L′:=L3(d−2;m1−1,...,m r−1) still satisfies the conditions of the theorem.Continuing like this until the residue class L′=L3(d′;m′1,...,m′r′)is such that r′≤8.For this class we then know that h1(L′)=0which gives us that h1(L)=0.4.Base point freenessTheorem4.1.Consider L=L3(d;m1,...,m r)on X r with m1≥m2≥···≥m r. Then L is base point free on X r if and only if the following conditions are satisfied(1)m r≥0,(2)d≥m1+m2and(2)4d≥m1+···m r+2if r≥8.4CINDY DE VOLDER AND ANTONIO LAFACEProof.First of all let us prove that the conditions are necessary.Obviously,if m r<0then m r E r⊂Bs(L);and if d<m1+m2then the strict transform of the line through P1and P2is contained in Bs(L).Now assume that(1)and(2)are satisfied,but4d≤m1+···m r+1and consider0L0(1) Proceeding as before,one can check that L⊗O Q r=L2(2d−m1;(d−m1)2, m2,...,m r)is standard and L2(2d−m1;(d−m1)2,m2,...,m r).(−K B r+1)=4d−m1−···−m r≤ing the results from[7],we obtain that L⊗O Q r has base points,which will also be base points of L.Now assume that all three conditions are satisfied(as before,we may even assume m r>0).If r≤8,the result follows from[6,Theorem6.2],so we assume that r≥9 and that the result holds for r′<r Consider the exact sequence(1).As before,one can see that L⊗O Q r=L2(2d−m1;(d−m1)2,m2,...,m r)is standard and since L2(2d−m1;(d−m1)2,m2,...,m r).K B r+1≤−2we know that L⊗O Q r is base point free(see[7,Lemma3.3(2)]).Also,because of theorem3.1,we know that h1(L3(d−2;m1−1,...,m r−1))=0,so L induces the complete linear system L⊗O Q r on Q r,which means in particular that L has no base points on Q r.On the other hand it is easily checked that L3(d−2;m1−1,...,m r−1)still satisfies the conditions of the theorem.Continue like this untill you have r′<r for the residue class(i.e. repeat this reasoning m r times).Denote L3(d−2m r;m1−m r,...,m r−1−m r)by L′.We then know that L′+mQ r⊂L.Since L has no base points on Q r,and since L′is base point free by our induction hypothesis,we obtain that Bs(L)=∅.5.Very amplenessTheorem5.1.Consider L=L3(d;m1,...,m r)on X r with m1≥m2≥···≥m r. Then L is very ample on X r if and only if the following conditions are satisfied(1)m r>0,(2)d≥m1+m2+1(d≥m1+1if r=1;d≥1if r=0)and(2)4d≥m1+···m r+3if r≥9.Proof.First of all let us note that the conditions are necessary.Obviously,if m r≤0 then L cannot separate on E r;and if d≤m1+m2then L cannot separate a zero-dimensional subscheme Z of length2of the strict transform of the line through P1 and P2.In case4d≤m1+···m r+2,one can see that L cannot separate Z if it is contained in D Q r.Now assume that all three conditions are satisfied.First of all consider the exact sequence0L0 Because of Theorem3.1we know that h1(L3(d;m1+1,m2,...,m r))=0,so Linduces the complete linear system L⊗O E1on E1.Since L⊗O E1=L2(m1),wesee that L separates on E1.Naturally,a similar reasoning can be done for any E i, so L separates on every E i.VERY AMPLENESS ON BLOWINGS-UP OF P35 Moreover,one can easily check that L(E i):=L3(d;m1,...,m i−1,m i+1,m i+1,..., m r)satisfies all the conditions of Theorem4.1,so that L(E i)is base point free on X r.This means that L can separate Z if∃i:Z∩E i=0but Z⊂E i. Combining the previous two results,we see that we now only need to show that L separates Z with Z∩E i=∅for all i=1,...,r.In case r=0,1or2,this is trivial.Now let us assume that r≥3and that the statement holds for r′<r.First look at the case where m r=1and consider the exact sequence 0L0 Proceeding as before,one can easily see that L⊗O Q r is standard.Moreover, 4d−m1−···−m r≥3(if r≥9this is condition(3)and if3≤r≤8this follows from(2)),so[8,Theorem2.1]implies that L⊗O Q r is very ample.Since h1(L3(d−2;m1−1,...,m r−1−1))=0(because of Theorem3.1)we then obtain that L separates on Q ing Theorem4.1we also obtain that L′:=L3(d−2;m1−1,...,m r−1−1,0)is base point free,which implies that L separates Z if Z∩Q r=∅.Let r′=max{i:m i>1}(or r′=0if all m i=1),then one can easily check that L′satisfies all the conditions of the theorem on X r′.So,using the induction hypothesis,we have that L′is very ample on X r′.But since Z on X r is disjoint with all E i,Z corresponds with a zero-dimensional subscheme on X r′(alsodisjoint with all E i).So may may conclude that L separates any Z.Now we assume m r>1and we assume that the statement holds for m′r<m r. Consider the exact sequence0L0 Proceeding similarly as for the case m r=1one can easily see that L separates all Z. Remark5.2.One can also use Theorem4.1and[1,Theorem2.1]to obtain a very ampleness result.However in this way only part of the complete class of very ample systems on X r are obtained.Remark5.3.Let A1,...,A r be general points on P3,let Y r be the blowing-up of P3along those r general points and let L3(d;m1,...,m r)(m1≥m2≥···≥m r) denote the complete linear system|dE0−m1E1−···−m r E r|on Y r.Since the very ampleness is an open property,Theorem5.1implies that L3(d;m1,...,m r)is very ample on Y r if m r>0,d≥m1+m2+1(d≥m1+1if r=1;d≥1if r=0)and 4d≥m1+···m r+3if r≥9.Of course the third condition will now no longer be a necessary condition.References[1]Mauro C.Beltrametti and Andrew J.Sommese.Notes on Embeddings of Blowups.J.Alg.,186:861–871,1996.[2]St´e phane Chauvin and Cindy De Volder.Some very ample and base point free linearsystems on generic rational surfaces.Math.Nachr.,245:45–66,2002.[3]Marc Coppens.Very ample linear systems on blowings-up at general points of projectivespaces.Canad.Math.Bull.,45(3):349–354,2002.[4]Marc Coppens.Very ample linear systems on blowings-up at general points of smoothprojective varieties.Pacific J.Math.,202(2):313–327,2002.6CINDY DE VOLDER AND ANTONIO LAFACE[5]Cindy De Volder and Antonio Laface.On linear systems of P3through multiple points.Preprint,math.AG/0311447,2003.[6]Cindy De Volder and Antonio Laface.Base locus of linear systems on the blowing-up ofP3along at most8general points.Preprint,math.AG/0401244,2004.[7]Brian plete linear systems on rational surfaces.Trans.Amer.Math.Soc.,289(1):213–226,1985.[8]Brian Harbourne.Very ample divisors on rational surfaces.Math.Ann.,272(1):139–153,1985.[9]Antonio Laface and Luca Ugaglia.On a class of special linear systems of P3.Preprint,math.AG/0311445,2003.Department of Pure Mathematics and Computeralgebra,Galglaan2,B-9000Ghent,BelgiumE-mail address:cdv@cage.ugent.beDipartimento di Matematica,Universit`a degli Studi di Milano,Via Saldini50,20100Milano,ItalyE-mail address:face@unimi.it。