Here we study three examples of differential geometric behaviour of projective varieties in positive characteristic: (1) the classification of smooth surfaces in P2n+1 whose m-th osculating spaces have everywhere dimension 2m (1 ≤ m ≤ n);

(2) hypersurfaces with Hessian rank 0; (3) singular hypersurfaces in weighted projective

spaces whose tangent sheaf is locally free and a subbundle of the restricted tangent

bundle

B. RUSSO

E. BALLICO

F. GIOVANETTI

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Rendiconti di Matematica, Serie VIIVolume 15, Roma (1995), 125-132Examples of di!erential geometric behaviour ofprojective varieties in positive characteristicE. BALLICO – F. GIOVANETTI – B. RUSSORiassunto: Studiamo tre esempi di proprietà di carattere geometrico-di!erenzialedelle varietà proiettive in caratteristica p: (1) classificazione di superfici in P2n+1 il cuim-esimo spazio osculatore ha sempre dimensione 2m (1 ' m ' n); (2) ipersuperficicon rango Hessiano 0; (3) ipersuperfici singolari di spazi proiettivi pesati con fasciotangente localmente libero.Abstract: Here we study three examples of di!erential geometric behaviour ofprojective varieties in positive characteristic: (1) the classification of smooth surfacesin P2n+1 whose m-th osculating spaces have everywhere dimension 2m (1 ' m ' n);(2) hypersurfaces with Hessian rank 0; (3) singular hypersurfaces in weighted projectivespaces whose tangent sheaf is locally free and a subbundle of the restricted tangentbundle.1 – IntroductionIn the last few years an active field of research was the study of pro-jective properties of subvarieties of Pn under the assumption that thealgebraically closed base field K has positive characteristic. In this paperwe give three independent results on this topic. In the second section weThis research was partially supported by MURST and GNSAGA of CNR (Italy).Key Words and Phrases: Positive characteristic – Osculating spaces – Hypersurfaces– Weighted projective space – Tangent sheaf – Rational scroll – Order of contact – Ruledsurface – Rational normal scroll – HyperosculationA.M.S. Classification: 14N05 – 14F10 – 14J70 – 14E25 – 14B10126 E. BALLICO – F. GIOVANETTI – B. RUSSO [2]show how to extend to positive characteristic the classification theoremin [6] on surfaces in P2n+1 with extremal osculating behaviour. Most ofsection three is devoted to a refinement of the non enumerative part of[18] about hypersurfaces with “pathological” (i.e. impossible in charac-teristic 0) di!erential geometric behaviour (see Theorems 3.1, 3.2, 3.3, 3.4and 3.5). The proofs of these five theorems (i.e. a reduction to the case ofplane curves) are simpler the the ones used in [18]. At the end of this sec-tion (just to give another example of funny behaviour of the derivativesin positive characteristic) we give the very easy extension of [4], Th. 0.1,from the case of hypersurfaces of Pn to the case of hypersurfaces of aweighted projective space (see Theorem 3.6). In each case we use freelybackground and proofs of the quoted references.2 – Surfaces with extremal osculating behaviourIn this section we show that only very minor changes (given below)will be su%cient to extend the theorem of [6] in positive characteristicunder the assumption that p := char(K) ! n, i.e. to prove the followingresult.Theorem 2.1. Assume p := char(K) ! n, and p ! 7 if n = 2.Let X . P2n+1, n ! 2, be a smooth projective surface not contained ina hyperplane and such that for every x " X and every m # n the m-thosculating space OscmX(x) at x has dimension 2m. Then X is a balancedrational normal scroll.The projective part of the proof of 2.1 is trivial, but the secondpart related to the classification of surfaces in positive characteristic isless trivial. The characteristic 0 proof of Theorem 2.1 given in [4] wasmainly based on the intermediate results and calculations made in [13]to prove particular cases of theorem. Everything in [13] used in [4] worksverbatim under our assumption on p until we arrive at [13], Corollary atpage 219, i.e. to the statement that X has Kodaira dimension 2(X) < 0and that X *= P2. It is very easy to check that X *= P2. Hence to find acontradiction we may assume 2(X) ! 0. Let S be the minimal model ofX and b the number of blowing ups of points needed to pass from S toX. Note that c2(X) = c2(S) + b. The proof of corollary works verbatim[3] Examples of di!erential geometric behaviour of etc. 127in positive characteristic if one can prove that c2(S) ! 0. By the positivecharacteristic classification of surfaces (see [8] or [7]) we have 2(X) *= 0.By [12], we have c2(X) ! 0 if 2(X) = 1. Hence we assume 2(X) = 2.Thus by [3], Th. A and 0.3.1, or [1] or [2], Th. A, a multiple of K + His still spanned by global sections and we have ([2], table at page 179)H(K+H) > 0 and K(K+H) > 0. Furthermore, since 2(X) = 2 we haveKH > 0. Since 2(X) = 2, by [9], Prop. 4.5, we have pc2(S)+ c1(S)2 ! 0.Since p ! n by assumption we still have (n $ 1)2c2(S) + nc1(S)2 ! 0.Hence if n ! 3 the numerical part of the proof of [13], Corollary at page219, works verbatim and we get a contradiction. Thus we may assumen = 2. For n = 2 we have c2(X) + 2(K + H)2 + 2KH + H2 = 0. Hencewe may assume c2(X) + K2 < 0, i.e. by Noether formula ((OX) < 0.By [15], Th. 8, we have p # 7. Hence we may assume p = 7, n = 2,c2(X) < 0. By [9], Prop. 4.5, we have p8c2(X)$ b9+(K2 + b) ! 0. Hence11c2(X)+2K2 ! 0. By [13], eq. (2), we have 11c2(X)+2K2+10KH = 0,contradiction. Now we need to look at the proofs in [4]. By the part ofthe proof of [13] after the corollary just extended to our setting we mayassume n ! 5, 2(X) < 0 and that X is not a relatively minimal model.Since 2(X) *= 2 we still have Bogomolov’s criterion of instability ([14],Th. 7). Hence we may use Reider’s method (see for instance [14], Cor. 8)to obtain the very ampleness of (K +H) except exactly the characteristic0 cases. Hence, using the very ampleness of K+H as in [4], the numericalpart of [4] at page 206 works verbatim and concludes the proof of 2.1.3 – Hypersurfaces with Hessian rank 0Here we extended the non enumerative part of [18] to the case inwhich the hypersurface has not too many singularities. In particular allthe results will be proven for every normal hypersurface; this result waspreviously known, since it is contained in the unpublished part of thethesis [10]. We stress that the proofs (i.e. the reduction to the case ofplane curves) are completely di!erent and much simpler that the proofs in[10], [16], [17] and [18]. For background and definitions (e.g. coordinategap number b2) see [18] and references therein. At the end of the sectionwe give an easy example (see Theorem 3.6) of extension to the case ofa weighted projective space of a positive characteristic funny behaviourknown for an ordinary projective space.128 E. BALLICO – F. GIOVANETTI – B. RUSSO [4]We fix an algebraically closed base field K with p := char(K) > 0.We fix homogeneous coordinates x0, . . . , xn on a projective space Pn; ifU is a homogeneous polynomial, let Ui or Di(U) (resp. Uij or Di,j(U)and so on) its partial derivative with respect to xi (resp. xi and xj).We fix an integral hypersurface X . Pn with degree d and let G beits homogeneous equation. We will give a total weight wtsg(X) for thecontributions of the singularities of X to give numerical bounds amongthe assumptions in all the theorems (except 3.6) proven in this section.Let {Sb}n$B be the family of all irreducible components of dimensionn $ 2 of Sing(X) (with its reduced structure); note that B = 3 if X isnormal. We take a general planeF . Pn and look at the integral curveC := X 2 F. For each singular point, P , of X let mP be the multiplicityof C and eP the multiplicity of the Jacobian ideal of C (i.e. the minimumof the intersection multiplicities at P of C with its partial derivative loci).Set(1) wtsg(X) := wtsg(C) =#P$Sing(C)8eP $ mP (mP $ 2)9.Now we give the statements of the theorems which will be provedhere.Theorem 3.1. Assume p > 2. Let X := {G = 0} be a degreed hypersurface of Pn with coordinate gap number b2(X) > 2. Assumewtsg(X) < d. Then the second order partial derivatives Gi,j of G vanishidentically.Theorem 3.2. Assume p > 2. Let X := {G = 0} be a degree dhypersurface of Pn with b2(X) = q = pe with e > 0. Assume Gi,j = 0 forall i, j and that for a general plane section C of X we have(2)#P$Sing(C)eP <81 $ (1/p)9d2 .The all the partial derivatives Dqi (G) of order q (0 # i # n) areidentically 0.Theorem 3.3. Assume p > 2. Let X := {G = 0} be a degree dhypersurface of Pn with b2(X) = q = pe with e > 0. Assume wtsg(X) <[5] Examples of di!erential geometric behaviour of etc. 129d. Then d = kq + 1 for some k and there are n + 1 degree k polynomialsP0, . . . , Pn such that(3) G =#ixiPqi .Now we consider the case p = 2.Theorem 3.4. Assume p = 2. Let X := {G = 0} be a degree dhypersurface of Pn with b2(X) = q = 2e with e ! 2. Assume that for ageneral plane section C of X we have#P$Sing(C)eP < (d/2) .The b2(X) = 2e with e ! 2 if and only if Gi,j = 0 for all i, j andDti(G) = 0 for all t with t = 2a and 1 # a < e.Theorem 3.5. Assume p = 2. Let X := {G = 0} be a degree dhypersurface of Pn with b2(X) = q = 2e with e ! 2. Assume that for ageneral plane section C of X we have#P$Sing(C)eP < (d/2) .Then d = kq +1 for some k and there are n+1 degree k polynomialsP0, . . . , Pn such thatG =#ixiPqi .Proof of 3.1 By linear algebra to show that G has all second orderpartial derivatives Gi,j identically 0, it is su%cient to show the corre-sponding vanishing in any system of coordinates and at a general point.Hence it is su%cient to restrict G to a general plane ". Then C := C 2"is non reflexive and 3.1 follows from [5], Th. 3.130 E. BALLICO – F. GIOVANETTI – B. RUSSO [6]By the last remark in [5] the bound in Theorem 3.1 can be improvedunder certain arithmetic conditions; for instance if p divides mP (resp.mP $ 1), then in the rigid hand side of (1) one can put mP2 (resp.mP (mP $ 1)) instead of mP (mP $ 2).Proof of 3.2 As in the previous proof we reduce to the case ofa general plane saection C := X 2 ". Note that the bound (2) is muchweaker than the bound in the statement of [11], Th. 5.5. Hence 3.2 followsquoting [5] instead of [11], Th. 5.1, in the proof of [11], Th. 5.5.Proof of 3.4 Reduce as in the previous proofs to the case of a planecurve and use [11], Th. 5.11.Proof of 3.5 Note that as in [18], proof of Th. 2.2 (or in [11], proofof Cor. 5.10) one obtains in a formal way the canonical form (3) as soonas one has proved the vanishing of all partial derivatives Gi,j and Dqi . Use3.2 if p < 2; use 3.4 and [11], Cor. 5.16, if p = 2In the last part of this paper we extend from the case of Pn to thecase of a weighted projective space W = P(w0, . . . , wn) the classificationtheorem [4], Th. 0.1, on singular hypersurface whose tangent sheaf is asubbundle of the restriction of TPn. Here we allow the case p = 0.Theorem 3.6. Let K be an algebraically closed field; set p :=char(K) ! 0. Let X be an integral hypersurface of a tame weightedprojective space W := P(w0, . . . , wn) (i.e. if p > 0 assume that all theweights of W are coprime to p). Let ! : Pn + W be the canonicalcover. Assume Sing(X) *= 3, !1(X) integral and not smooth and thatthe tangent sheaf TX is a subbundle of TW |X with OX(k) as quotientsheaf; assume that OW (k) is locally free in a neighborhood of X. Thenp > 0 and there are weighted homogeneous polynomials u, h, v such thatthe weighted homogeneous polynomial f of X is of the form(4) f = uph + vp .Viceversa, if p > 0, is given by (4) with k := deg(h) > 0 and OW (k)is invertible in a neighborhood of X := {f = 0}, Sing(X) *= 3 and at eachpoint of X one of the “weighted” partial derivatives of h does not vanish,then (TW |X)/TX 7= OX(k).[7] Examples of di!erential geometric behaviour of etc. 131The only problem to prove 3.6 is to find the “right” set up (tameweighted projective spaces and the local freeness of OW (k) in a neigh-borhood of X). After that, the proof of [4], Th. 0.1, works verbatim.REFERENCES[1] M. Andreatta – E. Ballico: On the adjuction over a surface in char. p,Manuscripta Math., 62 (1988), 227-244.[2] M. Andreatta – E. Ballico: Classification of projective surfaces with smallsectional genus: char. p $ 0, Rend. Sem. Mat. Univ. Padova, 84 (1990),175-193.[3] M. Andreatta – E. Ballico: On the adjuction process over a surface in char.p II: the singular case, J. reine agew. Math., 417 (1991), 77-85.[4] E. Ballico: The homogeneous equations of singular projective hypersurfaces withembedded tangent bundle, Rend. di Matem., 13 (1993), 461-465.[5] E. Ballico – A. Hefez: Non reflexive projective curves of low degree,ManuscriptaMath., 70 (1991), 385-396.[6] E. Ballico – R. Piene – H.S. Tai: A characterization of normal surface scrollsin term of their osculating spaces II , Math. Scand., 13 (1992), 204-206.[7] E. Bombieri: Methods of algebraic geometry in char. p and their applications,in: Algebraic Surfaces C.I.M.E., Cortona 1977, 57-96, Liguori, Napoli, (1981).[8] E. Bombieri – D. Mumford: Enriques’ classification of surfaces in char, p,II , in: Comples Analysis and Algebraic Geometry, 23-42, Cambridge UniversityPress, (1977).[9] T. Ekedhal: Foliations and inseparable morphisms, in: Proceedings of Symposiain Pure Math. 46, part. 2, 139-150, Providence, R.I.: Amer. Math. Soc., (1986).[10] A. Hefez: Duality for projective varieties, Ph. D. Thesis, M.I.T., (1985).[11] A. Hefez: Non reflexive curves, Compositio Math., 69 (1989), 3-35.[12] T. Katsura – K. Ueno: On elliptic surfaces in characteristic p, Math. Ann.,272 (1985), 291-330.[13] R. Piene – H.S. Tai: A characterization of balanced rational normal scrolls interms of their osculating spaces, in: Enumerative Geometry, Proc. Sitges (1987),215-224, Lect. Notes in Math. 1436, Springer-Verlag, (1990).[14] N.I. Shepherd-Barron: Unstable vector bundles and linear systems on surfacesin characteristic p, Invent. math., 106 (1991), 243-262.132 E. BALLICO – F. GIOVANETTI – B. RUSSO [8][15] N.I. Shepherd-Barron: Geography for surfaces of general type in positive char-acteristic, Invent. math., 106 (1991), 263-274.[16] M. Xu: Coordinate gap number and biduality , Acta Math. Sin., 32 (1989), 225-233.[17] M. Xu: The hyperosculating points of surfaces in P3, Compositio Math., 70(1989), 27-49.[18] M. Xu: The hyperosculating spaces of hypersurfaces, Math. Z., 211 (1992), 575-591.Lavoro pervenuto alla redazione il 4 maggio 1994modificato il 2 ottobre 1994ed accettato per la pubblicazione il 14 dicembre 1994.Bozze licenziate il 31 gennaio 1995INDIRIZZO DEGLI AUTORI:E. Ballico – F. Giovanetti – B. Russo – Dip. di Matematica – Università di Trento – 38050Povo (TN) – Italy – fax: Italy+461881624e-mail: ballico@ alpha.science.unitn.it – or – ballico@ itncisca.bitnete-mail: giovanetti@ alpha.science.unitn.it – or – giovanetti@ itncisca.bitnete-mail: russo@ alpha.science.unitn.it – or – russo@ itncisca.bitnet

Examples of differential geometric behaviour of projective varieties in positive characteristic

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Examples of differential geometric behaviour of projective varieties in positive characteristic

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