Seshadri constants and surfaces of minimal degree

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1 Seshadi constants and sufaces of minimal degee Wioletta Syzdek and Tomasz Szembeg Septembe 29, 2007 Abstact In [] we showed that if the multiple point Seshadi constants of an ample line bundle on a smooth pojective suface in vey geneal points satisfy cetain inequality then the suface is fibed by cuves computing these constants. Hee we chaacteize the bode case of polaized sufaces whose Seshadi constants in geneal points fulfill the equality instead of inequality and which ae not fibed by Seshadi cuves. It tuns out that these sufaces ae the pojective plane and sufaces of minimal degee. Intoduction and the main esult Given a smooth pojective vaiety X and a nef line bundle L on X, D ly defines the Seshadi constant of L at a point P X as the eal numbe ε(l; P ) := inf C L.C mult P C, whee the infimum is taken ove all educed and ieducible cuves passing though P (see [3] and [7, Chapt. 5]). This concept was extended by Xu [3] to finite subsets of a given vaiety. Let be an intege and P,..., P points in X. Then the -tuple Seshadi constant of L at the set P,..., P is the eal numbe ε(l; P,..., P ) := inf C {P,...,P } = L.C multpi C, whee the infimum is taken ove all ieducible cuves passing though at least one of the points P,..., P. Thee is an altenative and useful desciption of Seshadi constants in tems of the nef cone of a blown up vaiety. Specifically, let f : Y X be the blowing up of P,..., P X with exceptional divisos E,..., E. Then the Seshadi constant can be computed as { } ε(l; P,..., P ) = sup λ > 0 : f L λ E i is nef. The Kleiman citeion of ampleness implies then that the multiple point Seshadi constants ae subject to the following uppe bound which depends only on the degee of L and the numbe of points L ε(l; P,..., P ) dim X dim X =: α(l; ).

2 2 Wheneve thee is a stong inequality ε(l; P,..., P ) < α(l; ) () then the Seshadi constant is actually computed by a cuve and not appoximated by a sequence of cuves. Fo Seshadi constants at a single point this follows fom [, Lemma 5.2] and the agument easily modifies to the multiple point case. We call any cuve C with L.C ε(l; P,..., P ) = multpi C a Seshadi cuve fo L at the -tuple P,..., P. Oguiso (see [9]) studied the behavio of Seshadi constants ε(l; P ) unde the vaiation of the point P. He showed that the Seshadi function ε : X P ε(l; P ) R is semi-continuous and that it attains its maximal value at a set which is a complement of an at most countable union of Zaiski closed pope subsets of X i.e. fo a vey geneal point P. Oguiso aguments can be easily adapted to finite subsets. By ε(l; ) we will abbeviate the maximal value of the function ε : X (P,..., P ) ε(l; P,..., P ) R i.e. ε(l; ) := max ε. Nakamaye (see [8, Coollay 3]) obseved that in case of sufaces an inequality of type ε(l; ) < λ α(l; ) with a small facto λ has stong consequences fo the geomety of the suface. Namely thee exists a non-tivial fibation of X ove a cuve B whose fibes ae Seshadi cuves fo L. On sufaces this was studied in moe detail by Tutaj-Gasińska and the second autho [2]. Hwang and Keum passed fom sufaces to vaieties of abitay dimension (see [5]). In [] we stated eseach along the same lines fo multiple point Seshadi constants. In paticula we poved the following theoem. Theoem on fibations. Let X be a smooth pojective suface, L a nef and big line bundle on X and 2 a fixed intege. If ε(l; ) < α(l; ) (2) then thee exists a fibation f : X B ove a cuve B such that given P,..., P X vey geneal, fo abitay i =,..., the fibe f (f(p i )) computes ε(l; P,..., P ) i.e. the fibe is a Seshadi cuve of L. Futhemoe we showed that the bound in the Theoem is shap in the sense that fo evey intege thee exists a suface X togethe with an ample line bundle L such that one has equality in (2) and X is not fibed by Seshadi cuves of L. The pupose of this note is to chaacteize the pais (X, L) fo which one has an equality in (2) and X is not fibed by Seshadi cuves. The desciption of such pais is povided in the next theoem which is ou main esult.

3 3 Theoem Let 2 be a given intege, X a smooth pojective suface and L a nef and big line bundle on X such that ε(l; ) = α(l; ). If X is not fibed by Seshadi cuves fo L, then a) eithe = 2, X = P 2 and L = O(), b) o X is a suface of minimal degee in P and L = O X (). Remaks. (i) A simila theoem fo = was aleady obtained by us in [, Theoem 3.2] but the esult and the methods ae somewhat diffeent. (ii) A smooth suface is of minimal degee if and only if it is the Veonese suface in P 5 o a ational nomal scoll. This was poved by Del Pezzo (see [2]). Useful Lemmas Hee we ecall two Lemmas which ae essential fo the poof of the main esult. The fist Lemma goes back to Xu [3, Lemma ]. Lemma. Let X be a smooth pojective suface, let (C t, (P ) t,..., (P ) t ) t be a non-tivial one paamete family of pointed educed and ieducible cuves on X and let m i be positive integes such that mult (Pi ) t C t m i fo all i =,...,. Then fo = and m 2 C 2 t m (m ) + and fo 2 Ct 2 m2 i min{m,..., m }. The second lemma was obtained by Küchle in [6] and has puely aithmetical chaacte. Lemma.2 Let 2 and m,..., m Z be integes with m... m and m 2. Then we have ( ) 2 ( + ) m 2 i > m i + m ( + ). 2 Poof of the Theoem In this section we pove Theoem. Fist we give a shot oveview of the poof. Since the Seshadi constants of the line bundle in Theoem ae stictly less than the uppe bound, they must be computed by Seshadi cuves. We investigate popeties of these cuves in thee steps. Fist we show that unde assumptions of Theoem the multiplicities of Seshadi cuves in points P,..., P must all be equal to. This is an aithmetical pat of the poof. In the second step which is moe analytical, we show that Seshadi cuves must be ational. The thid step is geometical and ealizes Seshadi cuves as hypeplane sections of X embedded in a pojective space as a suface of minimal degee. Let us now tun to the details.

4 4 2. Multiplicities of Seshadi cuves By assumptions of the Theoem inequality () is satisfied so fo evey -tuple P,..., P thee exists a Seshadi cuve (C; P,..., P ). By [0, Poposition.3] thee ae finitely many such cuves fo evey -tuple. Fo a vey geneal -tuple we have the equality L.(C; P,..., P ) mult P i C = ( )L 2. (3) The numbe of algebaic families of cuves satisfying this equality is at most countable. So at least one of these families must not be discete. Fom now on we ae inteested in Seshadi cuves (C t ; (P ) t,..., (P ) t ) fo L moving in a non-tivial family ove some algebaic set. Let m i be the biggest integes such that mult (Pi ) t C t m i fo all t. Making a little bit smalle if necessay we may assume that actually m i = mult (Pi ) t C t fo all t. Renumbeing the points if necessay we may also assume that m... m. Thee ae the following thee cases possible: (A) m and m 2; (B) m =... m = ; (C) m = 0. In this step we want to exclude (A) and (C). In case (A) we ae in the position to apply Lemma.2. Thus ( ) 2 m i < + m 2 i m Ct 2, whee the second inequality is assued by Lemma.. Multiplying the above inequality by L 2 and applying the index theoem on the ight hand side we aive to the following inequality ( ) 2 m i L 2 < (L.C t ) 2. + Dividing by the sum of multiplicities and evoking (3) we obtain which is not possible. + L2 < 2 L 2

5 5 In case (C) if 3, then we have L.(C; P,..., P ) m i = L.(C; P,..., P ) m i = α(l; ) < 2 α(l; ). Hence ou Theoem on fibations shows that X is coveed by Seshadi cuves fo L contadicting the assumption of Theoem. If = 2, then by assumption we have ε(l; ) = 4 L2 and in this case we get the same contadiction by [2, Theoem]. Thus we showed that fo P,..., P vey geneal the Seshadi cuve fo L has multiplicities equal at all these points. In paticula we conclude fom (3) that Togethe with the index theoem we get L.(C; P,..., P ) = ( )L 2. (4) C 2. (5) 2.2 Rationality of Seshadi cuves In this pat we follow basically the defomation agument of [4] with necessay modifications. Fist we obseve that one can fix the points P,..., P and conside Seshadi cuves fo the -tuples P,, P, P with the last point moving. Among these cuves one can find again a non-tivial family (C t ; P,..., P, P t ) ove some smooth base. Fo t geneal the coesponding Kodaia-Spence map T t H 0 (C t, N Ct/X) factoizes in fact ove H 0 (C t, N Ct/X( P... P )). Lemma. implies that C 2 t. In view of (5) we obtain that in fact deg N Ct/X = C 2 t =. Since the image of the Kodaia-Spence map is non-zeo we conclude that the line bundle N Ct/X( P... P ) is tivial. Equivalently, thee is a section s in H 0 (C t, N Ct/X) whose zeo locus is exactly the diviso P P. Fixing P and moving instead anothe point in the tuple we get in the same manne sections s, s 2..., s in H 0 (C t, N Ct/X) whose zeo loci ae P P, P + P P,..., P P espectively. They ae obviously independent. This shows that N Ct/X is a line bundle of degee with at least sections. This can happen only in the case when C t is a ational cuve. Thus we showed that unde assumptions of Theoem the Seshadi cuves ae ational. 2.3 Embedding X as a suface of minimal degee It follows fom the last pat that X is ationally connected hence it is a ational suface. Since C 2 = fo Seshadi cuves, it follows fom the index theoem and (4) that the Seshadi cuves ae numeically equivalent. On ational sufaces this

6 6 implies the linea equivalence, so Seshadi cuves move in a single linea system. We call this system M and we show that M is in fact vey ample. Fist we show that M sepaates points. Let P and Q be two distinct points on X. Let C be a Seshadi cuve fo L lying in M and passing though P. It might happen that Q lies also on C. Taking P 2,..., P geneal on C we have that (C; P, P 2,..., P, Q) is a Seshadi cuve fo L. Taking Q vey geneal away of C thee exists also a Seshadi cuve (C ; P, P 2,..., P, Q ). Since C.C = this new cuve cannot pass though Q and thus we sepaated P and Q. Next we show that M sepaates tangent vectos. To this end fo a fixed point P it is enough to find two Seshadi cuves intesecting tansvesally at P. Again, this is the case fo the cuves C and C fom the agument above as they have = C.C points in common, so must intesect at evey of these points tansvesally. If = 2 then M has degee. This shows that X is P 2. Fo 3 and a smooth cuve C M we conside the exact sequence 0 O X O X (M) O C (C) 0. Since H (O X ) = 0 and h 0 (C, O C (C)) = as aleady established in the pevious pat, we conclude fom the long cohomology sequence that M has + sections. Hence the image of X unde the mapping given by M must be a suface of minimal degee. Acknowledgement. Duing the confeence Linea systems and subschemes held in Gent in Apil 2007 we pesented the esults of [] and began investigations which led to the pesent note. We would like to thank the oganizes of this confeence fo poviding a nice and stimulating atmosphee. We would also like to thank Bian Haboun fo nice discussions thee. The second autho was patially suppoted by KBN gant P03 A Refeences [] Baue, Th.: Seshadi constants on algebaic sufaces. Math. Ann. 33 (999), [2] Del Pezzo, P.: Sulle supeficie di odine n immese nello spazio di n + dimensioni. Rend. Cic. Mat. Palemo (886) [3] D ly, J.-P.: Singula Hemitian metics on positive line bundles. Complex algebaic vaieties (Bayeuth, 990), Lect. Notes Math. 507, Spinge-Velag, 992, pp [4] Ein, L., Lazasfeld, R.: Seshadi constants on smooth sufaces. In Jounées de Géométie Algébique d Osay (Osay, 992). Astéisque No. 28 (993), [5] Hwang, J.-M., Keum, J.: Seshadi-exceptional foliations. Math. Ann. 325 (2003), [6] Küchle, O.: Ample line bundles on blown up sufaces. Math. Ann. 304 (996), 5 55 [7] Lazasfeld, R.: Positivity in Algebaic Geomety I. Spinge-Velag, [8] Nakamaye, M.: Seshadi constants and the geomety of sufaces. J. Reine Angew. Math. 564 (2003), [9] Oguiso, K.: Seshadi constants in a family of sufaces. Math. Ann. 323 (2002),

7 7 [0] Syzdek, W.: Submaximal Riemann-Roch expected cuves and symplectic packing. Ann. Acad. Paedagog. Cac. Stud. Math. 6 (2007), 0 22 [] Syzdek, W., Szembeg, T.: Seshadi fibations of algebaic sufaces. axiv: v [math.ag], to appea in: Math. Nach. [2] Szembeg, T., Tutaj-Gasińska, H.: Seshadi fibations on algebaic sufaces, Ann. Acad. Paedagog. Cac. Stud. Math. 4 (2004), [3] Xu, G.: Ample line bundles on smooth sufaces. J. eine angew. Math. 469 (995), Wioletta Syzdek, Tomasz Szembeg Instytut Matematyki AP, ul. Podcho ażych 2, PL Kaków, Poland cuent addess: Mathematisches Institut, Univesität Duisbug-Essen, 457 Essen, Gemany syzdek@ap.kakow.pl szembeg@ap.kakow.pl