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The distribution of bidegrees of smooth surfaces in \(Gr(1,\mathbb{P}^ 3)\). (English) Zbl 0741.14017
The bidegree of a surface \(Y\subseteq Gr(1,\mathbb{P}^ 3)\) is the class of the surface in the codimension 2 Chow ring \(A^ 2Gr(1,\mathbb{P}^ 3)\cong\mathbb{Z} \eta\oplus\mathbb{Z} \eta'\), where \(\eta\) and \(\eta'\) are classes of the two families of planes in \(Gr(1,\mathbb{P}^ 3)\). In this paper we prove that if \(Y\subseteq Gr(1,\mathbb{P}^ 3)\) is a smooth surface of bidegree \((a,b)\), then if \(Y\) is not of general type, \(a\leq 3b\) and by symmetry, \(b\leq 3a\). If \(Y\) is of general type, then we show \(a\leq O(b^{4/3})\). Our method is to study the stability of the universal rank 2 bundle \({\mathcal E}\) on \(Gr(1,\mathbb{P}^ 3)\) restricted to \(Y\). I. Dolgachev and I. Reider have conjectured this restriction is semistable if \(Y\) is non-degnerate, and this implies \(a\leq 3b\). We show that if \({\mathcal E}\mid_ Y\) is unstable, then we get a strong bound on the hyperplane section genus of \(Y\), and this enables us to conclude our theorem.

14J25 Special surfaces
14M15 Grassmannians, Schubert varieties, flag manifolds
14J99 Surfaces and higher-dimensional varieties
Full Text: DOI EuDML
[1] Arbarello, E., Cornalba, M., Griffiths, P., Harris, J.: Geometry of algebraic curves. Berlin Heidelberg New York: Springer 1985 · Zbl 0559.14017
[2] Arrondo, E.: On congruences of lines in the projective space. Ph.D. Thesis, Universidad Complutense de Madrid, 1990
[3] Arrondo, E., Sols, I.: Classification of smooth congruences of low degree. J. Reine Angew. Math.393, 199-219 (1989) · Zbl 0649.14027
[4] Arrondo, E., Peskine, C., Sols, I.: Characterization of the spinor bundles by the vanishing of intermediate cohomology. (preprint)
[5] Barth, W., Peters, C., Van de Ven, A.: Compact complex surfaces. (Ergeb. Math. Grenzbeg., 3. Folge, vol. 4) Berlin Heidelberg New York: Springer 1984
[6] Bogomolov, F.: Holomorphic tensors and vector bundles on projective varieties. Math. USSR, Izv.13, 499-555 (1979) · Zbl 0439.14002 · doi:10.1070/IM1979v013n03ABEH002076
[7] Cossec, F., Dolgachev, I., Verra, A.: Unpublished manuscript
[8] Dolgachev, I., Reider, I.: On rank 2 vector bundles withc 1 2 =10 andc 2=3 on Enriques surfaces. (Preprint) · Zbl 0766.14031
[9] Eisenbud, D., Harris, J.: Curves in projective space. Montr?al: Les Presses de l’Universit? de Montr?al 1982
[10] Fano, G.: Nuove richerce sulle congrueze di Rette del 3? ordine prive di linea singolare. Mem. Accad. Sci. Torino, Cl. Sci. Fis. Mat. Nat.51, 1-80 (1902) · JFM 33.0685.02
[11] Fulton, W.: Intersection theory. (Ergeb. Math. Grenzgeb., 3. Folge, vol. 2) Berlin Heidelberg New York: Springer 1984 · Zbl 0541.14005
[12] Goldstein, N.: Scroll surfaces in Gr(1,P 3). In: Collono, A. et al. (eds.) Conference on algebraic varieties of small dimension. Turin, 1985. (Rend. Semin. Mat., Torino, Fasc. Spec., pp. 69-75) Torino: Editrice Univ. Levrotto & Bella 1986
[13] Griffiths, P., Harris, J.: Principles of algebraic geometry. New York: Wiley 1978 · Zbl 0408.14001
[14] Gross, M.: Surfaces in the four-dimensional Grassmannian. Ph.D. Thesis, U.C. Berkeley, 1990
[15] Gross, M.:Surfaces of bidegree (3,n) in Gr(1,P 3). (to appear 1990)
[16] Hartshorne, R.: Algebraic geometry. Berlin Heidelberg New York: Springer 1977 · Zbl 0367.14001
[17] Hartshorne, R., Rees, F., Thomas, E.: Nonsmoothing of algebraic cycles on Grassmann varieties. Bull. Am. Math. Soc.80 (no. 5), 847-851 (1974) · Zbl 0289.14011 · doi:10.1090/S0002-9904-1974-13537-8
[18] Hern?ndez, R., Sols, I.: Line congruences of low degree. In: Aroca, J.-M. et al., (eds.) G?om?trie alg?brique et applications. II. Singularit?s et g?om?trie complexe, pp. 141-154. Paris: 1987 Hermann
[19] Jessop, C. M.: A treatise on the line complex. Cambridge: Cambridge University Press 1903; reprinted Chelsea, Publishing Company 1969 · JFM 34.0702.06
[20] Kleiman, S.: Geometry on Grassmannians and applications to splitting bundles and smoothing cycles. Publ. Math., Inst. Hautes ?tud. Sci.36, 281-298 (1969) · Zbl 0208.48501 · doi:10.1007/BF02684605
[21] Miyaoka, Y.: The Chern classes and Kodaira dimension of a minimal variety. In: Oda, T. (ed.) Algebraic geometry. Sendai, 1985. (Adv. Stud. Pure Math., vol. 10, pp. 449-476 Tokyo: Kinokuniya 1987
[22] Mumford, D.: Geometric invariant theory. Berlin Heidelberg New York: Springer 1965; 2nd ed. 1982 · Zbl 0147.39304
[23] Pollack, A.: Codimension 2 subvarieties of Grassmannians. Ph.D. Thesis, Berkeley, 1978
[24] Ran, Z.: Surfaces of order 1 in Grassmannians. J. Reine Angew. Math.368, 119-126 (1986) · Zbl 0601.14042 · doi:10.1515/crll.1986.368.119
[25] Reid, M.: Bogomolov’s Theoremc 1 2 ?4c 2. In: Nagata, M. (ed.) Intl. Symp. on algebraic geometry, pp. 623-642. Kyoto. 1977. Tokyo: Kinokinuya 1978
[26] Roth, L.: Some properties of line congruences. Proc. Camb. Philos. Soc.26, 190-200 (1931) · JFM 57.0846.05 · doi:10.1017/S0305004100010306
[27] Verra, A.: Smooth surfaces of degree 9 inG(1, 3). Manuscr. Math.63, 417-435 (1988) · Zbl 0673.14026 · doi:10.1007/BF01357719
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