CLASSIFICATION OF ALGEBRAIC NON-RULED SURFACES WITH SECTIONAL GENUS LESS THAN OR EQUAL TO SIX

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1 E. L. Livorni Nagoya Math. J. Vol. (9), -9 CLASSIFICATION OF ALGEBRAIC NON-RULED SURFACES WITH SECTIONAL GENUS LESS THAN OR EQUAL TO SIX ELVIRA LAURA LIVORNI Introduction In this paper we have given a biholomorphic classification of smooth, connected, protective, non-ruled surfaces X with a smooth, connected, hyperplane section C relative to L, where L is a very ample line bundle on X, such that g = g(c) = g(l) is less than or equal to six. For a similar classification of rational surfaces with the same conditions see [Li]. From the adjunction formula 2g L (K x + L) where K x denotes the canonical line bundle on X. In the cases in which g =,, X is rational or ruled, see [Na]. Thus we have to examine only the cases in which g = 2,,. Our classification goes far beyond the birational classification of Leonard Roth [Ro] and the one that Polkin Ionescu [Io] has given for g <, Our main tool is the adjunction process and the results of Andrew John Sommese which are in [So]. Our notations are as in [So] except for the following. X will denote a smooth, connected, projective, non-ruled surface and L a very ample line bundle on X. Let L = K X L. Then h^(x) Φ g [So, (..2)], and L is generated by global sections [So, (.)], and the induced morphism φ has a 2-dimensional image [So, (2..) and (2.)]. Let X-+X be the Stein factorization and π: X-+X the induced morphism. Then Xis a non singular projective surface, π is the blow-up at a finite number of points of X, L = π*l and K ± + L are ample, and K x + L~π*(K x + L) [So, (2.3)]. We call (X, L) the minimal pair of (X, L). Our main goal is to classify the pairs (X, L) and eventually the pairs (X, L). Let d = (L 2 ), g = g(l\ d = (L 2 ), d' = (Z/ 2 ), Received June 3, 92. Revised December 9, 9.

2 so < o so d cl h\l) ' d' h\k x L) d h\l) cl λ XX) V>%X) (X,L) (X, L) 3 3 Λ (H t, <MD) Complete intersection of a quadric and a cubic. X = X,, -, One example is given by the complete intersection of three quadrics in P\ Blow up at most one point. 2 (H it <M)) x = x X = X. X is a minimal elliptic surface. 2 - X=rX. if3 surface blown up at one point if it exists. 9, 9, -A Λ X = X. X is a minimal elliptic surface if it exists. 2 K x is trivial, h\l\ e ) =, h\l\ e ) = X = X. X is an abelian surface. 9, 9,,,... -, -, K x is not trivial, h (L\c) =, Λ'( β ) = iγjf is not trivial, h\l\e) =, Λ'(L e) = K x is trivial, Λ (L e )=, AXLIί) = X = X. X is a hyperelliptic surface if it exists. Blow up at most one point. X is an Enriques surface. Blow up at most one point. Xis a K3 surface.

3 ALGEBRAIC NON-RULED SURFACES 3 g' g(k x + L), c\ = (K 2 χ), c\ = (K\). We fix the meaning of these symbols and the assumptions above throughout this paper. We have summarized our results in the above table, where using proposition (.) we have computed d f and g'. As one can see we have not been able to establish in all cases if there exists a surface satisfying the invariants that we have obtained. We are very grateful to Andrew J. Sommese whose suggestions have been very helpful. We would like to thank the referee for his careful revision.. Background material Since most of our background material is found in [So], we will mention here, without any details, only the results we need which cannot be found there. (.) PROPOSITION. Let L be a line bundle on a smooth, connected, projectίve surface X. Then: ) d' = g' + g-2 2) dd' < A(g - I) 2 3) d + d' = c\ + (g - ). See [Ro]. (.2) PROPOSITION. Let X be a smooth, connected, projective, surface embedded by a very ample line bundle S in P\ Then (.2.) JSf.J?(JS?.JSf - ) - (g(js - ) + 2X(χ) = 2K X K Z. See [Ha] pg. 3. (.3) PROPOSITION. Let X be a smooth, connected, projectίve surface and L a very ample line bundle on it. Suppose h (L) = 3, then: (X, L) = (X, L) = (fli, Fi (l)) = l, d =, c\ = 2) h\l) =, g =, Λ ι (X) =, h^(x) =, d =, cϊ = (X, L) = (X, ) = (H, (.) DEFINITION. A line bundle L on a projective variety is called spanned if h (L) is generated by the global sections. By [Ha, Lemma (.)] this is equivalent to say that h (L) is base-point-free.

4 ELVIRA LAURA LIVORNI (.) PROPOSITION. Let X be a smooth, connected, protective surface and L an ample line bundle on it If K x L<., then X is bίrational to Pj(C) cross a curve. Indeed since L is ample, one sees that H (K X ) = for all M> and that X is ruled by Castelnuovo's theorem [Bo + Hu, (3.)]. (.) PROPOSITION. With the notation and assumptions of the introduction, one has (i) d < d < 2g - 2, h\l)<h\l), (ii) d d = (Si) (cϊ) 2 is the number of points to blow up for the morphism X->X (hence in particular d = d(=}x = X), (iii) (2g dfld >c\>d-(2g- 2), (2g dfld >c\>d-(2g- 2), (iv) the following are equivalent: (iv. ) c\ = d - (2g - 2), (iv. 2) c\ = d - (2g - 2), (iv. 3) d = 2g - 2, (iv. ) K± «and X is a minimal model. (..) Remark. We note = (L iγr + L) - Φ) = 2g d hγ K x + L = π*(k x + L) and TΓ^L = L. Similarly one has (K x L)^2g-2-d. Proof. By (..) and (.), one has d < 2g - 2 because X is not ruled. The rest for (i) and (ii) follows from [So, (2.3)]. (iii): Since the argument is similar, we prove only the second assertion. By the algebraic index theorem, one has (L K x y>φ) (K x ), which reduces to (2g 2 d) 2 > d-c\. Since X is not ruled, one sees that \mk±\ Φ φ for some m >. Thus (K x -L + K±) > by ampleness of L + K±, which reduces to c\ > d - (2g - 2). (iv): Since K x + L = π*(k ± + L) and π*k x = K x, one has (K x -K x + L) = (K X K X + L), whence c\ + 2g d = c\ + 2g d as above. Thus (iv. l)φ(iv. 2). If (iv. 2) holds, then {K x -K x + L) = as above. Let

5 ALGEBRAIC NON-RULED SURFACES m > be such that \mk±\ Φ φ. Then (mk ± -K^ + L) = implies mk* ~ by ampleness of K± + L. Thus (iv. 2) implies (iv. ). (iv. ) implies (iv. 3) because (K± L) = 2g 2 d as above. If (iv. 3) holds, then one has c\ = by (iii), whence (iv. 2). q.e.d. (.) PROPOSITION. With the notation and assumptions of the introduction, assume that d = 2g 3. Then k{x) = and X is a minimal model {i.e. free from exceptional curves of the first kind). Proof claim: k{x) <. Assume that k{x) = 2, and let Y be its minimal model. Then by X{θ γ ) > and (K 2 Y) > [G + ΓJ, one has h (2K x ) = h (2K γ ) > l(2k γ ) - X(θ γ ) + (2K Y '2K Y - K γ )/2 = {Θ Y ) + (ίγ») > 2. Hence dim 2if^ >. Since L is very ample and L < r*l and πjl L, the linear system \L\ induces an embedding of X-(finite set). Since (2K^ L) = 2(2g d) = 2 (..), the generic number of \2K*\ is birationally sent to a line or a conic by the rational map associated to L. Thus X is ruled and this is a contradiction. Hence the claim is proved. claim: (iff) = and X is a minimal model. Since k{x) <, the minimal model Y of X satisfies (K γ ) =, whence (Kl) < (K γ ) =. By (.), (iii), one sees (UΓJ) =,-. By (.), (iv), one has (K±) Φ because (iv. 3) does not hold. Hence {Kl) = and the claim is proved. It now remains to prove K± ^. Indeed this follows from {K± L) = 2g-2-d = l (..). q.e.d.. Classification We use the notation and assumptions of the introduction. By (.3), one may assume h {L) >. By (.), one has d<d< 2g-2, < h {L) < h {L), and g < by our original assumption. We treat 2 cases d = 2g 2 and d < 2g 2, we can give comments on the existence later. (.) THEOREM. If d = 2g 2, then one of the following holds: (..) g =, d = d =, Λ (L) -, c x 2 =, X = X is an abelian surface or a hyperellίptic surface,

6 ELVIRA LAURA LIVORNI (..2) g =, d =, h\l) =, d = 9,, Λ (L) = d -, c^ =, X is α/ι Enrίques surface and L is very ample, (..3) g =,,, Λ (L) - +, A (L) = g + - (d ~ d\ (cΐ) =. X is a K3 surface and L is very ample. One has d = d and X = X is a complete intersection of a quadric and a cubic of P if g =, see [SaJ, and d = d or d- if g =,. Proof By (.), (iv), one sees that K x s and X is a minimal model, and one has X(θ±) =,, 2 [Bo + Hu]. Since if* «, one has H%L) = (/ > ) by Kodaira Vanishing Theorem. Hence one has ( * ) < h\l) < h\l) = l(θ x ) + < Z(^) + ^ -. Case i). X(# z ) =, X is an abelian surface or a hyperelliptic surface. In this case, one has g = and Λ (L) -, (#) = d -. Thus by (.2), one has d(d - ) - = 2(rf - ), whence d = and X = X This is (..). Case ii). {Θ X ), X is an Enriques surface. One has g > by (*), and h (L) = g. lΐ g =, then h (L) = and (XI) = d -, and one has d(d - ) = 2(d - ) by (.2), which has no solution. Thus this is a contradiction and one obtains g. Since X is an Enriques surface, one has h h (X) =, whence L is very ample by [So, (2.)]. Thus if d <, then H (L) is the subspace of H (L) with at least 2 base points [So, (2.3.2)] and h (L) < h (L) -2 =, which is a contradiction. Hence d 9 or. This is (..2). Case iii). l(θ x ) = 2, X is a i 3 surface. One has < Λ (L) < Λ (L) = g + < by (*). Since λ! (X) =, L is very ample [So, (2.)]. One has d = d and X is a complete intersection of a quadric and a cubic of P if g See [SaJ. Suppose now that d < d - 2. Then as in case ii), one has < h (L) < h (L) - 2 = and (K x ) = d -. By (.2), one has d(d - ) = 2(d - ). Thus d 2 d = which has no solution. Hence one has d > d. q.e.d. (.2) THEOREM. If d <2g 3, then one of the following holds: (.2.) =, d = d =, h\l) =, c\ =, Λ 2 ' (X) = 2, Λ! ' (X) -, IJSL^I is base-point-free and induces the elliptic fibratίon of X, (.2.2) # ==, d = d =, A (L) =, c 2 =, ίγ z consists of one conic

7 ALGEBRAIC NON-RULED SURFACES D and D is an exceptional curve of the first kind, its contraction is a minimal K3 surface, (.2.3) g =, d = d = 9, A (L) -, c\ =, /ι 2 ' (X) =, A^X) -,. X is α minimal elliptic surface. Proof Let C be a general member of \L\. We start with claim: \L\ C \ is a special linear system. If otherwise, one has < A (L) - < A (L C ) = l- + d< -2<. Thus h (L) =, =, d = d = 9, and X = X is an elliptic surface (, ). Hence by (.2), one has X(Φ Z ) =, that is 2X( Z ) =, which is a contradiction. Thus our claim is proved. Since L\ c is very ample and its degree d satisfies < d < άegk c, one has (**) 3 < h\l) - 2 < dim L c <d/2 < g - by Clifford's theorem [Sa 2, (.)]. Hence d >, g >. We claim that g=. Indeed if g =, then d = d = and A (L) = by (**), and one has (K 2 X) - by / > (K 2 X) > - by (.), (iii), (iv). Applying (.2), one has X(O Z ) =, i.e. V2X{Θ X ) = 2, which is a contradiction. Thus our claim g is proved, and we treat 2 cases h (L) > and h (L) =. Case i). A (L) >. In this case one has h (L) =, d - d - 9 by (**). Thus X = X is a minimal elliptic surface by (.). Since (K X L) = (..), Λ, 2 ' (X) =. Because otherwise jbγy contains a line which is a fiber of the elliptic fibration, which is a contradiction. One has h U (X) =, by X( A ) = >. Case ii). A (L) =. One has < d < d < 9, ( - dfjd > (K 2 X) > d - by (.), (iii), (iv), d(d - ) - + 2Z( X ) - 2{K\) by (.2). If d = 9, then (iζ ) = and X( Z ) = which is a contradiction. Thus d = or 9. Case ii.l) d =. In this case, > (K x ) > - and Z(^) = 2(K Z ). Thus (.SΓI) = - l, χ(φ x ) = 2. Hence ι 2 ' (X) > and l^ has an effective member Zλ Since (D-L) = 2 (..), D is a conic. Since X is not ruled, one sees dim D = and A 2 ' (X) = and h^{x) = by X(Θ X ) = 2. We claim that D is smooth and irreducible. Indeed by (D 2 ) =, D is reduced because

8 ELVIRA LAURA LIVORNI it is a conic. If D is reducible, then D A + D 2, where A and A are lines (A Φ A) and (D r D 2 ) =,. Since A ~ P, one has - 2 = 2p α (Z>) - 2 = (D '2JD + A), whence (Dr A) = and (A 2 ) = -. One has (A) = in the same way. This contradicts (D 2 ) =. Thus D is smooth and irreducible. Since (DL + if^) = >, there are no exceptional curve of the first kind collapsed by π and one has X X. Since D is an exceptional curve of the first kind, one can contract D by /: X-> Y. Then Y is a minimal KS surface by K γ ~ f*k x ~, Λ, M (Y) =. This is case (.2.2). Case ii.2). d =. In this case one has > (#) > - 2 and - + 2{Θ X ) = 2{K X ). Thus (K x ) =, Z( Z ) = 3. Hence Λ (-SLJ > 2, and ϋγ z contains a pencil of cubic curves by (K x -L) = 3 (..). Since X is not ruled, the general member of \K X \ must be a smooth elliptic curve. Thus one sees that \K X \ is a pencil free from base points by (K x ) =. Hence h (K x ) = 2 and h h (X) = by χ( z ) = 3. Since X is a minimal elliptic surface (k(x) = ), one has X = X. This is case (.2.). (.3) Remark. An example of an abelian surface in (..) is given in [Ho + Mu]. An example of (..2) with d = is given in [G + Ή. u p. 9] (V^w) given there is an Enriques surface of degree in P ) An example of (..3) with d = d and g = is given by a complete intersection of 3 quadrics of P. To construct an example of a surface in (..3) with d d and g, consider the quartic surface X in P 3. Let be a line on it and let J? be the line bundle (2 Z), which is very ample. Since l l = -2, = - Θ{2) = 2, we have. =.--2 =. Moreover, because K X = Θ Xo it follows that 2g(E) - 2 = E.(K Hι + E) = +. Hence g(i2) =. Furthermore, by the long cohomology sequence of the short exact sequence where Ce, we have that h (E) =, h\e) = = /ι 2 ( ). To construct an example of (.2.), let /: Y-+P be the P 2 -bundle P{Θ Θ{l) Θ(l)\ with

9 ALGEBRAIC NON-RULED SURFACES 9 the tautological line bundle L and let M = 3L + /*(). Then M is basepoint-free and its generic member X is smooth and K x ~ (f\ x )*Θ{ΐ). One can see that L is base-point-free and the induced map Y >F induced an embedding XcF in (.2.). We do not know if there are examples of hyperelliptic surfaces in (..), or examples of (..2), (.2.2) and (.2.3). REFERENCES [Bo+Hu] E. Bombieri and D. Husemoller, Classification and embeddings of surfaces, Proc. of Symposia in Pure Math., 29 (9), [G+Hi] P. A. Griffiths and J. Harris, Principles of Algebraic Geometry, A. Wiley- Inter science publication, (9). [Ha] R. Hartshorne, Algebraic Geometry, Springer-Verlag, New York, (9). [Ho+Mu] G. Horrocks and D. Mumford, Topology, Pergman Press, 2 (93), 3-. [Io] P. Ionescu, An enumeration of all smooth protective varieties of degree and, Preprint Series in Mathematics NR /9. [Li] E. L. Livorni, Classification of Algebraic surfaces with sectional genus less than or equal to six, I: Rational Surfaces, Pacific J. Math. Vol. 3, No. I, 9, 93-. [Na] M. Nagata, On rational surfaces I, Mem. Coll. Sci. Kyoto (A) 32 (9), 3-3. [Ro] L. Roth, On the protective classification of surfaces, Proc. London Math. Soc. 2nd series, 2 (93), 2-. [SaJ B. Saint-Donat, Protective models of K-3 surfaces. Amer. J. of Math., 9 (92), [Sa 2 ], On Petri's analysis of the linear system of quadrics through a canonical curve, Math. Ann., 2 (93), -. [So] A. J. Sommese, Hyperplane sections of projective surfaces I The adjunction mapping, Duke Math. J., (99) No. 2. Istituto dί Matematica Universita' Degli Studi DelΓAquila Degli Abruzzί via Roma, S3 UAQUILA Italy

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