East Asian mathematical journal

  • 제34권1호
  • /
  • Pages.21-28
  • /
  • 2018
  • /
  • 1226-6973(pISSN)
  • /
  • 2287-2833(eISSN)
영남수학회 (The Youngnam Mathematical Society)
권호 표지
DOI QR코드

DOI QR Code

DEFINING EQUATIONS OF RATIONAL CURVES IN SMOOTH QUADRIC SURFACE

  • LEE, Wanseok (Pukyong National University, Department of applied Mathematics) ;
  • Yang, Shuailing (Pukyong National University, Department of applied Mathematics)
  • 투고 : 2017.10.16
  • 심사 : 2018.01.02
  • 발행 : 2018.01.31
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

For a nondegenerate irreducible projective variety, it is a classical problem to study the defining equations of a variety with respect to the given embedding. In this paper we precisely determine the defining equations of certain types of rational curves in ${\mathbb{P}}^3$.

키워드

참고문헌

  1. M. Brodmann and E. Park, On varieties of almost minimal degree I: Secant loci of rational normal scrolls. J. Pure Appl. Algebra 214 (2010), 2033-2043 https://doi.org/10.1016/j.jpaa.2010.02.009
  2. M. Brodmann and E. Park, On varieties of almost minimal degree III: Tangent spaces and embedding scrolls. J. Pure Appl. Algebra 215 (2011), no. 12, 2859-2872 https://doi.org/10.1016/j.jpaa.2011.04.006
  3. M. Brodmann, E. Park and P. Schenzel, On varieties of almost minimal degree II: A rank-depth formula. Proc. Amer. Math. Soc. 139 (2011), no. 6, 2025-2032 https://doi.org/10.1090/S0002-9939-2010-10667-6
  4. M. Brodmann and P. Schenzel, Curves of Degree r+2 in $\mathbb{P}^r$:Cohomological, Geometric, and Homological Aspects. J. Algebra 242 (2001), 577-623 https://doi.org/10.1006/jabr.2001.8847
  5. M. Brodmann and P. Schenzel, On varieties of almost minimal degree in small codimension. J. Algebra 305 (2006), no.2, 789-801. https://doi.org/10.1016/j.jalgebra.2006.03.027
  6. M. Brodmann and P. Schenzel, Arithmetic properties of projective varieties of almost minimal degree. J. Algebraic Geometry 16 (2007), 347-400. https://doi.org/10.1090/S1056-3911-06-00442-5
  7. G. Castelnuovo, Sui multipli di une serie lineare di gruppi di punti appartenente ad une curva algebraic. Rend. Circ. Mat. Palermo (2) 7 (1893), 89-110. https://doi.org/10.1007/BF03012436
  8. M. Decker, G.M. Greuel and H. Schonemann, Singular 3 - 1 - 2 - A computer algebra system for polynomial computations. http://www.singular.uni-kl.de (2011).
  9. D. Eisenbud, M. Green, K. Hulek and S. Popescu, Restriction linear syzygies: algebra and geometry, Compositio Math. 141 (2005), 1460-1478. https://doi.org/10.1112/S0010437X05001776
  10. D. Eisenbud and S. Goto, Linear free resolutions and minimal multiplicity. Journal of Algebra 88 (1984) 89-133. https://doi.org/10.1016/0021-8693(84)90092-9
  11. La. Ein and R. Lazarsfeld, Syzygies and Koszul cohomology of smooth projective varieties of arbitrary dimension. Invention. Math. 111 (1993), 51-67. https://doi.org/10.1007/BF01231279
  12. T. Fujita, De ning equations for certain types of polarized varieties. in: Complex Analysis and Algebraic Geometry, Cambridge University Press, Cambridge (1977), 165-173.
  13. M. Green and R. Lazarsfeld, Some results on the syzygies of nite sets and algebraic curves. Compos. Math. 67 (1988), 301-314.
  14. L. Gruson, R. Lazarsfeld and C. Peskine, On a theorem of Castelnovo, and the equations de ning space curves . Inventiones mathematicae 72 (1983), 491-506. https://doi.org/10.1007/BF01398398
  15. J. Harris (with the collaboration of D. Eisenbud), Curves in Projective Space. Les Presses de l'Universite de Montreal, Montreal, 1982.
  16. L.T. Hoa, On minimal free resolutions of projective varieties of degree = codimension+2. J. Pure Appl. Algebra 87 (1993), 241-250. https://doi.org/10.1016/0022-4049(93)90112-7
  17. S. L'vovsky, On in ection points, monomial curves, and hypersurfaces containing projective curves. Math. Ann. 306 (1996), 719-735. https://doi.org/10.1007/BF01445273
  18. W. Lee and E. Park, On non-normal del Pezzo varieties. J. Algebra 387 (2013), 11-28 https://doi.org/10.1016/j.jalgebra.2013.04.013
  19. W. Lee and E. Park, Projective curves of degree=codimension+2 II. Internat. J. Algebra Comput. 26 (2016), no. 1, 95-104. https://doi.org/10.1142/S0218196716500041
  20. W. Lee and E. Park, On the minimal free resolution of curves of maximal regularity. Bull. Korean Math. Soc. 53 (2016), no. 6, 1707-1714. https://doi.org/10.4134/BKMS.b150890
  21. W. Lee, E. Park and P. Schenzel, On the classi cation of non-normal cubic hypersurfaces. J. Pure Appl. Algebra 215 (2011), 2034-2042. https://doi.org/10.1016/j.jpaa.2010.12.007
  22. W. Lee, E. Park and P. Schenzel, On the classi cation of non-normal complete intersection of two quadrics. J. Pure Appl. Algebra 216 (2012), no. 5, 1222-1234. https://doi.org/10.1016/j.jpaa.2011.12.009
  23. D. Mumford, Lectures on curves on an algebraic surface. With a section by G. M. Bergman. Annals of Mathematics Studies, No. 59 Princeton University Press, Princeton, N.J. 1966 xi+200 pp.
  24. U. Nagel, Arithmetically Buchsbaum divisors on varieties of minimal degree. Transactions of the American Mathematical Society 351, 4381-4409 (1999) https://doi.org/10.1090/S0002-9947-99-02357-0
  25. U. Nagel, Minimal free resolutions of projective subschemes of small degree. Syzygies and Hilbert functions, 209232, Lect. Notes Pure Appl. Math., 254, Chapman and Hall/CRC, Boca Raton, FL, 2007.
  26. E. Park, Projective curves of degree = codimension+2. Math. Z. 256 (2007), no. 3, 685- 697 https://doi.org/10.1007/s00209-007-0101-z
  27. E. Park, On hypersurfaces containing projective varieties. Forum Math. 27 (2015), no. 2, 843-875
  28. Schreyer,F-O, Syzygies of canonical curves and special linear series, Math.Ann.275,105-137 (1986). https://doi.org/10.1007/BF01458587
  29. Fyodor Zak, Determinants of projective varieties and their degrees. Algebraic transformation groups and algebraic varieties, 207-238, Encyclopaedia Math. Sci., 132, Springer, Berlin, 2004.