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Complete intersections on general hypersurfaces. (English) Zbl 1181.14057

The authors deal with the following natural question. Which complete intersections of small codimension are contained in a general hypersurface of degree \(d\) in \(\mathbb{P}^n\)? Let \(a_1,\dots, a_r\) be natural numbers less than a number \(d\). One wishes to know whether a complete intersection of type \((a_1,\dots,a_r)\) is contained in a general hypersurface of degree \(d\) in the projective space. The authors show using Terracini’s lemma that this is true if and only if for general forms \(F_i\) of degrees \(a_i\) and \(G_i\) of degree \(d-a_i\), the ideal generated by the \(F\)’s and \(G\)’s contain all forms of degree \(d\). In particular, \(2r\geq n+1\). They restrict their attention thus to the two cases \(2r=n+1\) and \(2r=n+2\) and classify all possibilities when this will occur.

MSC:

14N05 Projective techniques in algebraic geometry
14M10 Complete intersections
14M07 Low codimension problems in algebraic geometry
13A02 Graded rings
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[1] B. Å dlandsvik, Varieties with an extremal number of degenerate higher secant varieties, J. Reine Angew. Math. 392 (1988), 16–26. · Zbl 0649.14029 · doi:10.1515/crll.1988.392.16
[2] D. Anick, Thin algebras of embedding dimension three, J. Algebra 100 (1986), 235–259. · Zbl 0588.13013 · doi:10.1016/0021-8693(86)90076-1
[3] E. Arrondo and L. Costa, Vector bundles on Fano 3-folds without intermediate cohomology, Comm. Algebra 28 (2000), 3899–3911. · Zbl 1004.14010 · doi:10.1080/00927870008827064
[4] E. Carlini, Codimension one decompositions and Chow varieties, Projective varieties with unexpected properties (C. Ciliberto et al., eds.), pp. 67–79, de Gruyter, Berlin, 2005. · Zbl 1101.14066
[5] ——, Binary decompositions and varieties of sums of binaries, J. Pure Appl. Algebra 204 (2006), 380–388. · Zbl 1081.14073 · doi:10.1016/j.jpaa.2005.05.008
[6] M. Catalano-Johnson, The possible dimensions of the higher secant varieties, Amer. J. Math. 118 (1996), 355–361. · Zbl 0871.14043 · doi:10.1353/ajm.1996.0012
[7] M. V. Catalisano, A. V. Geramita, and A. Gimigliano, Ranks of tensors, secant varieties of Segre varieties and fat points, Linear Algebra Appl. 355 (2002), 263–285. · Zbl 1059.14061 · doi:10.1016/S0024-3795(02)00352-X
[8] ——, Erratum to: “Ranks of tensors, secant varieties of Segre varieties and fat points”, Linear Algebra Appl. 367 (2003), 347–348. · Zbl 1073.14550 · doi:10.1016/S0024-3795(03)00455-5
[9] L. Chiantini and C. Ciliberto, On the concept of \(k\) -secant order of a variety, J. London Math. Soc. (2) 73 (2006), 436–454. · Zbl 1101.14067 · doi:10.1112/S0024610706022630
[10] J. V. Chipalkatti, Decomposable ternary cubics, Experiment. Math. 11 (2002), 69–80. · Zbl 1046.14500 · doi:10.1080/10586458.2002.10504469
[11] C. Ciliberto, Geometric aspects of polynomial interpolation in more variables and of Waring’s problem, European Congress of Mathematics, vol. I (Barcelona, 2000), Progr. Math., 201, pp. 289–316, Birkhäuser, Basel, 2001. · Zbl 1078.14534
[12] R. Fröberg, An inequality for Hilbert series of graded algebras, Math. Scand. 56 (1985), 117–144. · Zbl 0582.13007
[13] A. V. Geramita, Inverse systems of fat points: Waring’s problem, secant varieties of Veronese varieties and parameter spaces for Gorenstein ideals, The curves seminar at Queen’s, vol. X (Kingston, 1995), Queen’s Papers in Pure and Appl. Math., 102, pp. 2–114, Queen’s Univ., Kingston, ON, 1996. · Zbl 0864.14031
[14] A. V. Geramita and H. K. Schenck, Fat points, inverse systems, and piecewise polynomial functions, J. Algebra 204 (1998), 116–128. · Zbl 0934.13013 · doi:10.1006/jabr.1997.7361
[15] P. Griffiths and J. Harris, On the Noether–Lefschetz theorem and some remarks on codimension-two cycles, Math. Ann. 271 (1985), 31–51. · Zbl 0552.14011 · doi:10.1007/BF01455794
[16] A. Grothendieck, Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA 2), Doc. Math. (Paris), 4, Soc. Math. France, Paris, 2005. · Zbl 1079.14001
[17] J. Harris, Algebraic geometry, A first course, Grad. Texts in Math., 133, Springer-Verlag, New York, 1992. · Zbl 0779.14001
[18] M. Hochster and D. Laksov, The linear syzygies of generic forms, Comm. Algebra 15 (1987), 227–239. · Zbl 0619.13007 · doi:10.1080/00927872.1987.10487449
[19] A. Iarrobino and V. Kanev, Power sums, Gorenstein algebras, and determinantal loci, Lecture Notes in Math., 1721, Springer-Verlag, Berlin, 1999. · Zbl 0942.14026 · doi:10.1007/BFb0093426
[20] H. P. Kley, Rigid curves in complete intersection Calabi–Yau threefolds, Compositio Math. 123 (2000), 185–208. · Zbl 1054.14054 · doi:10.1023/A:1002012414149
[21] J. M. Landsberg and L. Manivel, On the ideals of secant varieties of Segre varieties, Found. Comput. Math. 4 (2004), 397–422. · Zbl 1068.14068 · doi:10.1007/s10208-003-0115-9
[22] S. Lefschetz, On certain numerical invariants of algebraic varieties with application to abelian varieties, Trans. Amer. Math. Soc. 22 (1921), 327–406. JSTOR: · JFM 48.0428.03 · doi:10.2307/1988897
[23] C. Madonna, Rank-two vector bundles on general quartic hypersurfaces in \(\mathbb P^4,\) Rev. Mat. Complut. 13 (2000), 287–301. · Zbl 0981.14019 · doi:10.5209/rev_REMA.2000.v13.n2.17073
[24] C. Mammana, Sulla varietà delle curve algebriche piane spezzate in un dato modo, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 8 (1954), 53–75. · Zbl 0055.39003
[25] J. Migliore and R. M. Miró-Roig, Ideals of general forms and the ubiquity of the weak Lefschetz property, J. Pure Appl. Algebra 182 (2003), 79–107. · Zbl 1041.13011 · doi:10.1016/S0022-4049(02)00314-6
[26] N. Mohan Kumar, A. P. Rao, and G. V. Ravindra, Four-by-four Pfaffians, Rend. Sem. Mat. Univ. Politec. Torino 64 (2006), 471–477. · Zbl 1183.14060
[27] F. Severi, Una proprieta’ delle forme algebriche prive di punti muiltipli, Rend. Accad. Lincei 2 (1906), 691–696. · JFM 37.0131.01
[28] R. P. Stanley, Weyl groups, the hard Lefschetz theorem, and the Sperner property, SIAM J. Algebraic Discrete Methods 1 (1980), 168–184. · Zbl 0502.05004 · doi:10.1137/0601021
[29] E. Szabó, Complete intersection subvarieties of general hypersurfaces, Pacific J. Math. 175 (1996), 271–294. · Zbl 0888.14018
[30] J. Watanabe, The Dilworth number of Artinian rings and finite posets with rank function, Commutative algebra and combinatorics (Kyoto, 1985), Adv. Stud. Pure Math., 11, pp. 303–312, North-Holland, Amsterdam, 1987. · Zbl 0648.13010
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