Codimension two nonsingular subvarieties of quadrics scrolls and classification in degree <i>d</i>≤10

Mark Andrea A. de CATALDO

doi:10.2969/jmsj/05040879

Codimension two nonsingular subvarieties of quadrics

scrolls and classification in degree d≤10

Mark Andrea A. de CATALDO

Author information

Keywords: Classification, liaison, low codimension, low degree, quadric, scroll, vector bundle

JOURNAL FREE ACCESS

1998 Volume 50 Issue 4 Pages 879-902

DOI https://doi.org/10.2969/jmsj/05040879

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Published: 1998 Received: November 28, 1996 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: October 20, 2006 Reason for correction: - Correction: TITLE Details: Wrong : Codimension two nonsingular subvarieties of quadrics: scrolls and classification in degree d≤10
Right : Codimension two nonsingular subvarieties of quadrics

Date of correction: October 20, 2006 Reason for correction: - Correction: SUBTITLE Details: Right : scrolls and classification in degree d≤10

Date of correction: October 20, 2006 Reason for correction: - Correction: ABSTRACT Details: Wrong : Let X be a codimension two nonsingular subvariety of a nonsingular quadric _??_n of dimension n≥5. We classify such subvarieties when they are scrolls. We also classify them when the degree d≤10. Both results were known when n=4.
Right : Let X be a codimension two nonsingular subvariety of a nonsingular quadric 2ⁿ of dimension n≥5. We classify such subvarieties when they are scrolls. We also classify them when the degree d≤10. Both results were known when n=4.

Date of correction: October 20, 2006 Reason for correction: - Correction: KEYWORD Details: Right : Classification, liaison, low codimension, low degree, quadric, scroll, vector bundle

Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) V. Ancona, G. Ottaviani, “Some Applications of Beilinson's Theorem to Projective Spaces and Quadrics,” Forum Math. 3 (1991), 157-176.
2) E. Arbarello, M. Cornalba, P. A. Griffiths, J. Harris, Geometry of Algebraic Curves, Vol. I, Grundlehren Math. Wiss. 267, Springer-Verlag, New York, (1985).
3) E. Arrondo, I. Sols, “Classification of smooth congruences of low degree,” J. Reine Angew. Math. 393 (1989), 199-219.
4) E. Arrondo, I. Sols, “On congruences of lines in the Projective Space,” Soc. Mat. de France, Mémorie n⁰ 50, Suppl. au Bull. de la S.M.F., Tome 120, 1992, fascicule 3.
5) W. Barth, Submanifolds of low codimension in projective space,” in Proc. Intern. Cong. Math., Vancouver (1974), 409-413.
6) M. Beltrametti and A. J. Sommese, The adjunction theory of complex projective varieties, Expositions in Mathematics, 16 (1995), 398+xxi pages, Walter De Gruyter, Berlin.
7) R. Braun, G. Ottaviani, M. Schneider. F.-O. Schreyer, “Boundedness of nongeneral type 3-folds in P⁵,” in Complex Analysis and Geometry, ed. by V. Ancona and A. Silva, 311-338 (1993), Plenum Press, New York.
8) M. A. A. de Cataldo, “The genus of curves on the three dimensional quadric,” Nagoya Math. J. 147 (1997), 193-211.
9) M. A. A, de Cataldo, “A finiteness theorem for low-codimensional nonsingular subvarieties of quadrics,” Trans. A.M.S. 349 (6) (1997), 2359-2370.
10) M. A. A. de Cataldo, “Some adjunction-theoretic properties of codimension two nonsingular subvarieties of quadrics,” Canadian. J. of Math. 49 (4) (1997), 675-695.
11) M. A. A. de Cataldo, Ph.D Thesis, Notre Dame (1995).
12) M. L. Fania, L. E. Livorni, “Degree ten manifolds of dimension n≥3,” Math. Nachr. 188 (1997), 79-108.
13) T. Fujita, “On polarized manifolds whose adjoint bundles are not semipositive,” in Algebraic Geometry, Sendai 1985, Adv. Stud. Pure Math. 10 (1987), 167-178.
14) W. Fulton, Intersection Theory, Ergeb. Math. Grenzgeb. (3) 2, Springer-Verlag, Berlin, (1984).
15) N. Goldstein, “Scroll surfaces in Gr(1, P³),” Conference on Algebraic Varieties of Small Dimension (Turin 1985), Rend. Sem. Mat. Univ. Polit. Special Issue (1987), 69-75.
16) M. Gross, “Surfaces of degree 10 in the Grassmannian of lines in 3-space,” J. Reine Angew. Math. 436 (1993), 87-127.
17) J. Harris, Curves in Projective Space, with the collaboration of D. Eisenbud, Les Presses de L' Université de Montréal (1982).
18) R. Hartshorne, Algebraic Geometry, Graduate Texts in Math. 52, Springer-Verlag, New York, (1978).
19) W. V. D. Hodge, D. Pedoe, Methods of Algebraic Geometry, Vol. 1 (1953), Vol. 2 (1995), Vol. 3 (1954), Cambridge University Press.
20) P. Ionescu, “Embedded projective varieties of small invariants,” in Proceedings of the 1982 Week of Algebraic Geometry, Bucharest, Lecture Notes in Math. 1056 (1984), 142-187, Springer-Verlag, New York.
21) P. Ionescu, “Embedded projective varieties of small invariants, III,” in Algebraic Geometry, Proceedings of Conference on Hyperplane Sections, L'Aquila, Italy, 1988, ed. by A. J. Sommese, A. Biancofiore, L. E. Livorni, Lecture Notes in Math., 1417 (1990), 138-154, Springer-Verlag, New York.
22) S. Kleiman, “Geometry on Grassmannians and applications to splitting bundles and smoothing cycles,” Publ. Math. IHES, 36, (1969), 281-297.
23) C. Okonek, “Notes on Varieties of Codimension 3 in P^N,” Man. Math. 84, 421-442, (1994).
24) C. Okonek, M. Schneider, H. Spindler, Vector Bundles on Complex Projective Spaces, Progr. Math. 3 (1980), Birkhäuser, Boston.
25) G. Ottaviani, “On Cayley Bundles on the Five-Dimensional Quadric,” Bollettino U.M.I. (7) 4-A (1990), 87-100.
26) G. Ottaviani “On threefolds in P⁵ which are scrolls,” Ann. Scuola, Norm. Sup. Pisa Cl. Sci. Ser. (4) 19 (1992), 451-471.
27) B. Shiffman, A. J. Sommese, Vanishing theorems on complex manifolds, Progr. Math. 56 (1985) Birkhä user, Boston.
28) A. J. Sommese, A. Van de Ven, “Homotopy groups of pullbacks of varieties,” Nagoya Math. J., Vol. 102 (1986), 79-90.

Right : [1] V. Ancona, G. Ottaviani, “Some Applications of Beilinson's Theorem to Projective Spaces and Quadrics,” Forum Math. 3 (1991), 157-176.
[2] E. Arbarello, M. Cornalba, P. A. Griffiths, J. Harris, Geometry of Algebraic Curves, Vol. I, Grundlehren Math. Wiss. 267, Springer-Verlag, New York, (1985).
[3] E. Arrondo, I. Sols, “Classification of smooth congruences of low degree,” J. Reine Angew. Math. 393 (1989), 199-219.
[4] E. Arrondo, I. Sols, “On congruences of lines in the Projective Space,” Soc. Mat. de France, Mémorie n⁰ 50, Suppl. au Bull. de la S. M. F., Tome 120, 1992, fascicule 3.
[5] W. Barth, Submanifolds of low codimension in projective space,” in Proc. Intern. Cong. Math., Vancouver (1974), 409-413.
[6] M. Beltrametti and A. J. Sommese, The adjunction theory of complex projective varieties, Expositions in Mathematics, 16 (1995), 398+xxi pages, Walter De Gruyter, Berlin.
[7] R. Braun, G. Ottaviani, M. Schneider. F. -O. Schreyer, “Boundedness of nongeneral type 3-folds in P⁵,” in Complex Analysis and Geometry, ed. by V. Ancona and A. Silva, 311-338 (1993), Plenum Press, New York.
[8] M. A. A. de Cataldo, “The genus of curves on the three dimensional quadric,” Nagoya Math. J. 147 (1997), 193-211.
[9] M. A. A, de Cataldo, “A finiteness theorem for low-codimensional nonsingular subvarieties of quadrics,” Trans. A. M. S. 349 (6) (1997), 2359-2370.
[10] M. A. A. de Cataldo, “Some adjunction-theoretic properties of codimension two nonsingular subvarieties of quadrics,” Canadian. J. of Math. 49 (4) (1997), 675-695.
[11] M. A. A. de Cataldo, Ph. D Thesis, Notre Dame (1995).
[12] M. L. Fania, L. E. Livorni, “Degree ten manifolds of dimension n≥3,” Math. Nachr. 188 (1997), 79-108.
[13] T. Fujita, “On polarized manifolds whose adjoint bundles are not semipositive,” in Algebraic Geometry, Sendai 1985, Adv. Stud. Pure Math. 10 (1987), 167-178.
[14] W. Fulton, Intersection Theory, Ergeb. Math. Grenzgeb. (3) 2, Springer-Verlag, Berlin, (1984).
[15] N. Goldstein, “Scroll surfaces in Gr (1, P³),” Conference on Algebraic Varieties of Small Dimension (Turin 1985), Rend. Sem. Mat. Univ. Polit. Special Issue (1987), 69-75.
[16] M. Gross, “Surfaces of degree 10 in the Grassmannian of lines in 3-space,” J. Reine Angew. Math. 436 (1993), 87-127.
[17] J. Harris, Curves in Projective Space, with the collaboration of D. Eisenbud, Les Presses de L' Université de Montréal (1982).
[18] R. Hartshorne, Algebraic Geometry, Graduate Texts in Math. 52, Springer-Verlag, New York, (1978).
[19] W. V. D. Hodge, D. Pedoe, Methods of Algebraic Geometry, Vol. 1 (1953), Vol. 2 (1995), Vol. 3 (1954), Cambridge University Press.
[20] P. Ionescu, “Embedded projective varieties of small invariants,” in Proceedings of the 1982 Week of Algebraic Geometry, Bucharest, Lecture Notes in Math. 1056 (1984), 142-187, Springer-Verlag, New York.
[21] P. Ionescu, “Embedded projective varieties of small invariants, III,” in Algebraic Geometry, Proceedings of Conference on Hyperplane Sections, L'Aquila, Italy, 1988, ed. by A. J. Sommese, A. Biancofiore, L. E. Livorni, Lecture Notes in Math., 1417 (1990), 138-154, Springer-Verlag, New York.
[22] S. Kleiman, “Geometry on Grassmannians and applications to splitting bundles and smoothing cycles,” Publ. Math. IHES, 36, (1969), 281-297.
[23] C. Okonek, “Notes on Varieties of Codimension 3 in P^N,” Man. Math. 84, 421-442, (1994).
[24] C. Okonek, M. Schneider, H. Spindler, Vector Bundles on Complex Projective Spaces, Progr. Math. 3 (1980), Birkhäuser, Boston.
[25] G. Ottaviani, “On Cayley Bundles on the Five-Dimensional Quadric,” Bollettino U. M. I. (7) 4-A (1990), 87-100.
[26] G. Ottaviani “On threefolds in P⁵ which are scrolls,” Ann. Scuola, Norm. Sup. Pisa Cl. Sci. Ser. (4) 19 (1992), 451-471.
[27] B. Shiffman, A. J. Sommese, Vanishing theorems on complex manifolds, Progr. Math. 56 (1985) Birkhä user, Boston.
[28] A. J. Sommese, A. Van de Ven, “Homotopy groups of pullbacks of varieties,” Nagoya Math. J., Vol. 102 (1986), 79-90.

Date of correction: October 20, 2006 Reason for correction: - Correction: PDF FILE Details: -

Abstract

Let X be a codimension two nonsingular subvariety of a nonsingular quadric 2ⁿ of dimension n≥5. We classify such subvarieties when they are scrolls. We also classify them when the degree d≤10. Both results were known when n=4.

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