General blowups of ruled surfaces. and higher order embeddings. T. Szemberg, H. Tutaj-Gasinska. Abstract

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1 Geneal blowups of uled sufaces and highe ode embeddings T. Szembeg, H. Tutaj-Gasinska PREPRINT 1999/16 Abstact In this note we study blowups of algebaic sufaces of Kodaia dimension =?1 at geneal points and thei appopiate highe ode embeddings. Intoduction Let X be a smooth pojective vaiety and L an ample line bundle on X. In the ecent yeas seveal authos studied line bundles of the fom M = L? E i ; whee : e X?! X is the blowing up at points P1 ; : : : ; P X with exceptional divisos E 1 ; : : : ; E. Fo P 1 ; : : : ; P geneal and X a suface Kuchle [15] investigated when M is ample. D'Almeida, Hischowitz [10] and Coppens [9] studied when M is vey ample fo P 1 ; : : : ; P geneal and X a suface o P n. On the othe hand seveal concepts of highe ode embeddings have been intoduced and studied by Beltametti, Fancia, Sommese and othes (see e.g. [3], [6]). Beltametti and Sommese [7] studied when M denes an embedding of ode k fo any choice of points P 1 ; : : :; P. The aim of the pesent pape is to genealize the above esults and study when M is k?vey ample fo geneal points on sufaces with Kodaia dimension =?1. A st step in this diection, fo X = P, was done in [16]. The main esults of this pape is Theoem 11. The poof builds upon the genealization of Reide's theoem due to Beltametti, Fancia and Sommese [3]. Also Seshadi constants of line bundles tun out to be a new and useful tool in this set-up Mathematics Subject Classication: 14E5, 14C0

2 Notation and backgound mateial We wok thoughout ove the eld C of complex numbes. All vaieties ae assumed to be smooth and pojective. If X is a vaiety, by K X we denote the canonical diviso of X, and by H i (X; F) = H i (F) the cohomology goups of a coheent sheaf F on X. Fo line bundles L and divisos D on X we use exchangeably the notation L + D, O X (D) L o O X (D + L). The numeical equivalence is denoted by. The Hodge Index Theoem is abbeviated by HIT and used on a suface Y in the fom of inequality H D (HD) fo divisos H; D Y with H > 0. Fo the pupose of ou pape we intoduce the following teminology: a line bundle L on a smooth pojective vaiety X is said to be k?vey ample at a 0?dimensional subscheme Z of length k + 1 if the estiction mapping is sujective. H 0 (X; L)?! H 0 (X; L O Z ) We ecall that L is k?vey ample if it is k?vey ample at evey 0?dimensional subscheme Z of length k + 1. Note that a line bundle is 0?vey ample i it is globally geneated and 1?vey ample i it is vey ample. If : e X?! X is a blowing up as in the Intoduction we say that a subscheme Z e X is admissible if the length of its estiction to evey exceptional diviso is. Assuming = = 1 in the denition of M one is foced to estict attention to admissible subschemes. Indeed, we have deg M = 1 hence thee is no hope Ei to sepaate longe subschemes contained in the exceptional divisos. A line bundle which is k?vey ample at admissible subschemes is still geometically meaningful. In paticula thee ae no (k + 1)?secant (k? 1)?planes in the embedding of X e besides those containing the lines E i of couse. Moeove it denes a biational mapping of the Hilbet scheme S [k+1] X of 0-dimensional subschemes of X of length k + 1 into Gassmanian Gass(k + 1; H 0 (X; L) ), cf. [8]. The stuctue of this mapping is inteesting on its own and will be studied elsewhee. We ecall the following citeion fo a line bundle to be k?vey ample on a suface. This genealization of Reide's theoem to the k?vey ample case was poved by Beltametti and Sommese in []. Poposition 1 Let X be a smooth suface and let L be a nef line bundle on X, such that L 4k + 5. Then eithe K X + L is k-vey ample o thee exists an eective diviso D satisfying the following conditions: (1) L? D is Q-eective, i.e. thee exists a positive intege m such that jm(l? D)j 6= ;. () D contains a subscheme Z of the length k + 1, such that the mapping is not sujective. (3) LD? k? 1 D < LD < k + 1: H 0 (K X L)?! H 0 (K X L O Z ) We shall make fequent use of the following elementay inequality without futhe notice. Lemma Fo any integes m 1 ; :::; m m i! and the equality holds i all m i ae equal. m i ;

3 3 Seshadi constants In the sequel we shall make use of Seshadi constants of a line bundle. Let us ecall thei denitions and some popeties. Let L be an ample line bundle on a smooth pojective vaiety X and let x X be a point. Seshadi constant of L at x is the eal numbe "(L; x) = supf" such that L? "E is nefg; whee : e X?! X is the blowing up of X at x with the exceptional diviso E. Moe geneally, fo paiwise distinct points x 1 ; : : : ; x in X the multiple point Seshadi constant of L at x 1 ; : : : ; x is the numbe "(L; x 1 ; : : :; x ) = supf" such that L? " E i is nefg; whee is the obvious blowing up of X. Fo vey geneal points we wite shotly "(L; ) and fo geneal points: " (L; ). Of couse "(L; ) " (L; ). On the othe hand we ecall (cf. [11, Lemma 1.4]) that " (L; ) "(L; )? fo any positive. Modifying slightly the agument of Ein and Lazasfeld in [1] we obtain the following Poposition 3 Let L be an ample line bundle on a smooth pojective suface S. Let > 1 be an intege. If L? + 1 and L:G fo all ieducible cuves G on S except possibly G 1 ; : : : ; G t, then "(L; x) fo all but nitely many points x X n fg 1 [ : : :[ G t g. In paticula " (L; 1) and "(L; 1). Fo eade's convenience we ecall also the following citeion due to Kuchle [15]. Poposition 4 Let L be a nef line bundle on a smooth pojective n?fold. Fo we have ( np p ) L n n "(L; ) min "(L; 1); ; L n (? 1) n?1 : Remak 5 We note that in the case of sufaces p? 1 p "(L; ) min "(L; 1); L ; since the eal valued function p?1 is deceasing fo. Ruled sufaces Now, we ecall some facts about uled sufaces, cf. [13, Pop. V..0 and Pop. V..1]. Lemma 6 Let S be a uled suface ove a cuve C 0 of genus g, with a section C, a be f, and invaiant e :=?C. If G S ac + bf is an ieducible cuve on S, dieent fom C and f then

4 4 a > 0; b ae fo e 0, if e < 0 then eithe a = 1; b 0 o a ; b 1 ae. The next lemma was poved in [1, Pop. 3.3 and Pop. 3.4] fo k?spanned line bundles, but the poof applies without any changes to the k?vey ample case. Lemma 7 Let S be a uled suface ove a cuve C 0 of genus g > 0, with a section C, a be f and the invaiant e. Let d be a positive intege. And let M S be a line bundle such that M S ac + bf. 1. If e 0 and a d and b ae + g? + maxfd + ; eg then the line bundle M S is d-vey ample.. If e < 0 and a d and b ae + g + d () then the line bundle M S is d-vey ample. Remak 8 Fom the poof of the above Lemma in [1] it follows that the line bundle L S = M S? K S is nef and L S (d + ). In the case of Hizebuch sufaces we have the following chaacteization. Lemma 9 Let L = ac + bf be a line bundle on the Hizebuch suface F. Then L is k?vey ample if and only if a k and b a + k. Main esult and poof Fist we fomulate the following geneal esult which follows immediately fom the poof of Theoem 3.1 in [9]. Poposition 10 Let S be a smooth pojective suface and M S a vey ample line bundle on S. Let : X?! S be the blowing up of S at s = h 0 (M)? 3 geneal points P 1 ; : : : ; P s. Then the line bundle is globally geneated. M = M S? Of couse this esult is optimal. Hence, we assume in the sequel that k 1. Let S be a minimal suface of Kodaia dimension (S) =?1. The case of S = P was studied in [16] so we estict ou attention to uled sufaces. Theoem 11 Let S be a minimal uled suface ove a cuve C 0 of genus g, with a section C, a be f and the invaiant e. Let k be a positive intege and d an intege satisfying d 4k. Suppose that M S ac + bf is a line bundle satisfying with the above d the assumption () o () of Lemma 7. Let be a positive intege such that 5 E i (MS? K S )? (4k + 5) : Let : X?! S be the blowing up of X at P 1 ; : : : ; P. The line bundle M = M S? is k?vey ample at evey admissible subscheme Z of X fo P 1 ; : : :; P geneal. E i ()

5 5 Poof The poof goes in seveal steps, the details ae technically involved but the idea should be tanspaent. We st check that Poposition 1 can be applied in ou situation, this involves Seshadi constants. Consideing vaious cases we show then that a diviso poduced by the Reide agument cannot exist povided Z is admissible. We conside L S = P M S? K S = (a + )C + (b? g + + e)f and L := M? K X. Thus L = L S? E i. Claim 1. The line bundle L is ample. Equivalently it is enough to show that " (L S ; ) >. To this end it is enough to show that "(L S ; ) >. Heeto we apply Poposition 4. Fist we veify that "(L S ; 1) >. Fom Remak 8 (and assumption k 1) we have L S (d + ) 7 > 5: Let G S be an ieducible cuve. If G = f then L S :f = a + : If G = C and e 0 then it may happen that L S :G = 0, in case e < 0 we get L S C 9. If G pc + qf is an ieducible cuve dieent fom C and f then L S :G =?(a + )pe + q(a + ) + p(b + + e? g): If e 0 then fom () and Lemma 6 we get L S :G?(a + )pe + pe(a + ) + p(ae + maxfd + ; eg + e) 6: Similaly, if e < 0 then eithe p = 1 and q 0 and L S G 9 o p and q 1 pe and L S :G p(d + ) 1: Fom Poposition 3 with = and G 1 = C we obtain thus "(L S ; 1) > : Now, the condition p L S (? 1) > follows fom ou estiction on. Indeed since the function p?1 is deceasing it is enough to check the condition in the exteme case = 5 (L S? (4k + 5)), which is elementay. Thus Poposition 4 implies that L is ample. Claim. The condition L 4k + 5 is obviously satised. Claim 3. Suppose now that M = L + K X is not k-vey ample at an admissible subscheme Z X. Then, accoding to Poposition 1 thee exists a diviso D satisfying, in paticula, the condition (3) of Poposition 1. We will show that this is not possible. Fist, we obseve that if D is a Reide diviso then only the case L:D? k? 1 D 0 (1) can occu. Indeed, fom ou estiction on and Remak 8 we get L = L S? 4 11 (16k + 36k + 9): We distinguish two cases

6 6 a) k + LD k + 1 and b) 0 < L:D k + 1: In the st case we use HIT and aive at 11 (16k + 36k + 9)(LD? k? 1) (LD) : Elementay calculus shows that this neve holds fo k 1 and LD in the given ange. In the case b) HIT gives a contadiction at once. Claim 4. Let D S be a diviso on S such that D S 0. Then L S D S 3k + 3: () Since L S is nef it is enough to pove the claim fo D S ieducible. Assume that e 0. If D S = f then L S f = a + 4k + 3k + 3. If D S C then e = 0 and L S C = maxf4k + ; eg 3k + 3. If D S pc + qf with p > 0 and q pe then L S D S = (q? pe)(a + ) + p(b? g + + e) (4k + 1)e + maxf4k + + ; eg 3k + 3: The poof fo e < 0 is simila. We poceed now studying the situation case by case accoding to numeical popeties of D and its undelying diviso D S. Fist we give an oveview of the cases. They ae undestood up to enumbeing the points P 1 ; : : : ; P. 1. D = P E i with m i 0,. D = D 0 + P i=s+1 m ie i whee D 0 = D S? P s k ie i and D S is nontivial, 0 k i mult Pi D S, m i 0 and m s+1 1,.1. (D 0 ) 0 then we go to Case 3. with D 0,.. (D 0 ) > 0 and..1. s > 0 and k 1 1,... s = 0 3. D = D S? P E i with 0 k i mult Pi D S, hee we omit the assumption Z D, 3.1. D = 0, 3.. D < 0 and D S 0, < D S 1k, k < D S. Case 1. P Fist we exclude the possibility that D is suppoted in the exceptional divisos i.e. D = m ie i with m i 0. Renumbeing the points P 1 ; : : : ; P P we may assume s that m 1 : : : m s > m s+1 = : : : = m = 0 with s 1 i.e. D = m ie i with m i > 0. Then LD = m i s and D =? m i?s: Since Z is admissible we have k + 1 = h 0 (O Z ) s. This and (1) imply 0 s? (k + 1) LD? (k + 1) D?s;

7 7 a contadiction. Case. Now, let D be of the fom D = D S? k i E i + i=s+1 m i E i Pwith D S nontivial, 0 k i mult Pi D S, m i 0 and m s+1 1. Let D 0 = D S? s k ie i. If (D 0 ) 0 then D 0 satises (1) and we poceed as in the Case 3. Othewise we study two subcases. Case..1 Assume that s > 0 and k 1 1. Then we have 0 < (D 0 ) = DS? which implies P s k i D S > k i 1: Futhe k + 1 LD = LD 0 + P i=s+1 m i LD 0 + and consequently L S D S k? 1 + Applying HIT on S, Lemma and estictions on we get L S This gives k i < L S D S (L S D S ) (k? 1) + 4(k (L S? (4k + 5))) L S k i : k i < 11(k? 1) + (k? 47) Recalling that L S (4k + ) we see that the above inequality cannot be satised. Case.. If all k i in the above ae equal 0 we have D 0 = D S. In paticula D S > 0. Since L SD S k + 1 also in this case, HIT gives an immediate contadiction. Case 3. Now, we assume that D P P = D S? E i with 0 k i mult Pi D S. Case 3.1 If D = 0 then DS = k i : k i : k i. We have also L S:D S = L:D? P k i P k + 1? k i. Combining this with HIT on S, Lemma and ou assumptions on we get X X D S L S (L S :D S ) (k + 1)? 4(k + 1) k i + 4( k i ) (k +1) +4(k +1) This is equivalent to k i +4 which togethe with Remak 8 yields k i (k +1) +4(k +1)D S (L S?(4k +5))D S : D S L S 11(k + 1) + (4k? 6)D S ; (3D S? 11)k + (8D S? )k + 14D S? 11 0: Fo k 0 this inequality holds only if D S = 0 i.e. all k i = 0. In this case we have L S D S k + 1: If S is uled then D S must be of the fom D S = pf o D S = pc + ep f fo some positive intege p. Intesecting in both cases D S with L S we get a contadiction with the above inequality.

8 8 Case 3..1 Since D < 0 fom (1) we get k LD P P = L S D S? k i. As DS 0 we can use (). Putting these togethe we have k i k + 3. Now, (1) gives LD? (k + 1) D S? k i? Thus LD < 0 contadicting L being ample. Case 3.. Fom (1) we have in paticula L being ample. It follows that Thus L S D S? k i? LD? D k + 1 and? k D = D S? k i? D S L S D S? k i + k i?k? 3 : k i k i? D S = LD? D k + 1: k i D S + k and L S D S k + D S + 1: Since DS > 0 we have fom HIT and Remak 8 This gives (4k + ) D S L S D S (k + D S + 1) : 0 4(D S)? 4(8k + 6k + 1)D S + (k + 1) : Elementay calculus shows that this neve holds fo k 1 and DS in the given ange (the oots of the undelying quadatic equation being < 1 and > 1k). Case 3..3 Since D?k P P by (1), we obtain k i? k D S. As befoe LD k, hence L S D S k + k i. Taking the above into account and applying HIT on S we get L ( S k i? k) k + 4k k + 10 i 11 (L S? (4k + 5)) k i : This gives L S( k i? 11k) 11k + (4k? 50) Since the coecient at L S is positive we get by Remak 8 (3k + 8k + 58) Now, D < 0 implies 1k < D S < P k i, thus k i : k i 11k(3k + 33k + 8): 3k? 7k < 0; a contadiction. Acknowledgments. This note was witten duing st autho's stay at the Max Planck Institute. It is a pleasue to thank the Institute fo poviding excellent woking conditions and atmosphee. The st autho was patially suppoted by KBN gant P03A

9 9 Refeences [1] Beltametti, M.C., Sommese, A.J.: On k-spannedness fo pojective sufaces. LNM 1417, 4-51 (1988) [] Beltametti, M.C., Sommese, A.J.: Zeo cycles and k-th ode embeddings. Pojective sufaces and thei classication, Symp. Math., INDAM 3, 33-48, Academic Pess 1988 [3] Beltametti, M.C., Fancia, P., Sommese, A.J.: On Reide's method and highe ode embeddings. Duke Math. J. 58, (1989), [4] Beltametti, M.C., Sommese, A.J.: On the pesevation of k?vey ampleness unde adjunction. Math. Z. 1, (1993) [5] Beltametti, M. C., Sommese, A. J.: Pojective sufaces with k?vey ample line bundles of degee 4k + 4, Nagoya Math. J. 136 (1994), [6] Beltametti, M.C., Sommese, A.J.: The Adjunction Theoy of Complex Pojective Vaieties. Expositions in Mathematics, 16, Walte de Guyte, Belin (1995). [7] Beltametti, M.C., Sommese, A.J.: Notes on embeddings of blowups. J. Algeba 186, (1996), [8] Catanese, F., Gottsche, L.: d-vey ample line bundles and embeddings of Hilbet schemes of 0-cycles. Manuscipta Mat. 68, (1990) [9] Coppens, M., Embeddings of geneal blowing-ups at points, J. Reine Angew. Math. 469 (1995), [10] D'Almeida, J., Hischowitz, A.: Quelques plongements pojectifs non speciaux de sufaces ationelles. Math. Z. 11, (199) [11] Ein, L., Kuchle, O., Lazasfeld, R., Local positivity of ample line bundles, J. Di. Geom. 4 (1995), 193{19 [1] Ein, L., Lazasfeld, R., Seshadi constants on smooth sufaces. Jounes de Geometie Algebique d'osay (Osay, 199). Asteisque 18 (1993), 177{186 [13] Hatshone, R.: Algebaic geomety. Spinge 1977 [14] Kuchle, O.: Ample line bundles on blown up sufaces. Math. Ann. 304, (1996) [15] Kuchle, O., Multiple point Seshadi constants and the dimension of adjoint linea seies, Ann. Inst. Fouie 46, (1996), [16] Tutaj-Gasinska, H.: Embeddings of blown-up plane. To appea in Geom. Dedicata Tomasz Szembeg Max Planck Institut fu Mathematik P.O. Box 780, D-5307 Bonn, Gemany szembeg@mpim-bonn.mpg.de Halszka Tutaj-Gasinska Instytut Matematyki, Uniwesytet Jagiellonski, Reymonta 4, PL Kakow, Poland htutaj@im.uj.edu.pl