A construction of codimension three arithmetically Gorenstein subschemes of projective space

Migliore, Juan; Peterson, Chris

doi:10.1090/S0002-9947-97-01978-8

A construction of codimension three arithmetically Gorenstein subschemes of projective space
HTML articles powered by AMS MathViewer

by Juan C. Migliore and Chris Peterson PDF

Trans. Amer. Math. Soc. 349 (1997), 3803-3821 Request permission

Abstract:

This paper presents a construction method for a class of codimension three arithmetically Gorenstein subschemes of projective space. These schemes are obtained from degeneracy loci of sections of certain specially constructed rank three reflexive sheaves. In contrast to the structure theorem of Buchsbaum and Eisenbud, we cannot obtain every arithmetically Gorenstein codimension three subscheme by our method. However, certain geometric applications are facilitated by the geometric aspect of this construction, and we discuss several examples of this in the final section.

References

Edoardo Ballico, Giorgio Bolondi, and Juan Carlos Migliore, The Lazarsfeld-Rao problem for liaison classes of two-codimensional subschemes of $\textbf {P}^n$, Amer. J. Math. 113 (1991), no. 1, 117–128. MR 1087803, DOI 10.2307/2374823
D. Bayer and M. Stillman, Macaulay, a computer system for computing in Commutative Algebra and Algebraic Geometry.
M. Beltrametti, M. Schneider, and A. J. Sommese, Threefolds of degree $9$ and $10$ in $\textbf {P}^5$, Math. Ann. 288 (1990), no. 3, 413–444. MR 1079870, DOI 10.1007/BF01444540
Mauro Beltrametti, Michael Schneider, and Andrew J. Sommese, Threefolds of degree $11$ in $\textbf {P}^5$, Complex projective geometry (Trieste, 1989/Bergen, 1989) London Math. Soc. Lecture Note Ser., vol. 179, Cambridge Univ. Press, Cambridge, 1992, pp. 59–80. MR 1201375, DOI 10.1017/CBO9780511662652.006
Giorgio Bolondi, Surfaces in $\mathbf P^4$ and deficiency modules, Classification of algebraic varieties (L’Aquila, 1992) Contemp. Math., vol. 162, Amer. Math. Soc., Providence, RI, 1994, pp. 49–63. MR 1272693, DOI 10.1090/conm/162/01527
Giorgio Bolondi and Juan Carlos Migliore, Configurations of linear projective subvarieties, Algebraic curves and projective geometry (Trento, 1988) Lecture Notes in Math., vol. 1389, Springer, Berlin, 1989, pp. 19–31. MR 1023387, DOI 10.1007/BFb0085921
Giorgio Bolondi and Juan Carlos Migliore, The Lazarsfeld-Rao property on an arithmetically Gorenstein variety, Manuscripta Math. 78 (1993), no. 4, 347–368. MR 1208646, DOI 10.1007/BF02599318
Robert Braun, Giorgio Ottaviani, Michael Schneider, and Frank-Olaf Schreyer, Boundedness for nongeneral-type $3$-folds in $\textbf {P}_5$, Complex analysis and geometry, Univ. Ser. Math., Plenum, New York, 1993, pp. 311–338. MR 1211889
David A. Buchsbaum and David Eisenbud, Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension $3$, Amer. J. Math. 99 (1977), no. 3, 447–485. MR 453723, DOI 10.2307/2373926
E. D. Davis, A. V. Geramita, and F. Orecchia, Gorenstein algebras and the Cayley-Bacharach theorem, Proc. Amer. Math. Soc. 93 (1985), no. 4, 593–597. MR 776185, DOI 10.1090/S0002-9939-1985-0776185-6
Wolfram Decker, Lawrence Ein, and Frank-Olaf Schreyer, Construction of surfaces in $\textbf {P}_4$, J. Algebraic Geom. 2 (1993), no. 2, 185–237. MR 1203684
Wolfram Decker and Sorin Popescu, On surfaces in $\textbf {P}^4$ and $3$-folds in $\textbf {P}^5$, Vector bundles in algebraic geometry (Durham, 1993) London Math. Soc. Lecture Note Ser., vol. 208, Cambridge Univ. Press, Cambridge, 1995, pp. 69–100. MR 1338413, DOI 10.1017/CBO9780511569319.004
Geir Ellingsrud and Christian Peskine, Sur les surfaces lisses de $\textbf {P}_4$, Invent. Math. 95 (1989), no. 1, 1–11 (French). MR 969410, DOI 10.1007/BF01394141
E. Graham Evans and Phillip Griffith, The syzygy problem, Ann. of Math. (2) 114 (1981), no. 2, 323–333. MR 632842, DOI 10.2307/1971296
A. V. Geramita and J. C. Migliore, Hyperplane sections of a smooth curve in $\textbf {P}^3$, Comm. Algebra 17 (1989), no. 12, 3129–3164. MR 1030613, DOI 10.1080/00927878908823898
A.V. Geramita and J. Migliore, Reduced Gorenstein Codimension Three Subschemes of Projective Space, Proc. Amer. Math. Soc. 125 (1997), 943–950.
Phillip Griffiths and Joseph Harris, Residues and zero-cycles on algebraic varieties, Ann. of Math. (2) 108 (1978), no. 3, 461–505. MR 512429, DOI 10.2307/1971184
L. Gruson, R. Lazarsfeld, and C. Peskine, On a theorem of Castelnuovo, and the equations defining space curves, Invent. Math. 72 (1983), no. 3, 491–506. MR 704401, DOI 10.1007/BF01398398
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
Robin Hartshorne, Stable reflexive sheaves, Math. Ann. 254 (1980), no. 2, 121–176. MR 597077, DOI 10.1007/BF01467074
—, Families of Curves in $\mathbb {P}^{3}$ and Zeuthen’s Problem, Mem. Amer. Math. Soc. (to appear).
C. Huneke, Numerical invariants of liaison classes, Invent. Math. 75 (1984), no. 2, 301–325. MR 732549, DOI 10.1007/BF01388567
Craig Huneke and Bernd Ulrich, General hyperplane sections of algebraic varieties, J. Algebraic Geom. 2 (1993), no. 3, 487–505. MR 1211996
Martin Kreuzer, On $0$-dimensional complete intersections, Math. Ann. 292 (1992), no. 1, 43–58. MR 1141784, DOI 10.1007/BF01444608
Martin Kreuzer, Vektorbündel und der Satz von Cayley-Bacharach, Regensburger Mathematische Schriften [Regensburg Mathematical Publications], vol. 21, Universität Regensburg, Fachbereich Mathematik, Regensburg, 1989 (German). Dissertation, Universität Regensburg, Regensburg, 1989. MR 1027141
Ernst Kunz, Almost complete intersections are not Gorenstein rings, J. Algebra 28 (1974), 111–115. MR 330158, DOI 10.1016/0021-8693(74)90025-8
R. Maggioni and A. Ragusa, The Hilbert function of generic plane sections of curves of $\textbf {P}^3$, Invent. Math. 91 (1988), no. 2, 253–258. MR 922800, DOI 10.1007/BF01389367
Mireille Martin-Deschamps and Daniel Perrin, Sur la classification des courbes gauches, Astérisque 184-185 (1990), 208 (French). MR 1073438
Rosa M. Miró-Roig, The degree of smooth non-arithmetically Cohen-Macaulay threefolds in $\textbf {P}^5$, Proc. Amer. Math. Soc. 110 (1990), no. 2, 311–313. MR 1031451, DOI 10.1090/S0002-9939-1990-1031451-6
J. Migliore, Curves and their Hyperplane Sections, lecture notes for the Notre Dame Algebraic Geometry Seminar (1989).
Juan C. Migliore, An introduction to deficiency modules and liaison theory for subschemes of projective space, Lecture Notes Series, vol. 24, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1994. MR 1301773
Christian Okonek, Notes on varieties of codimension $3$ in $\mathbf P^N$, Manuscripta Math. 84 (1994), no. 3-4, 421–442. MR 1291131, DOI 10.1007/BF02567467
Christian Okonek, Michael Schneider, and Heinz Spindler, Vector bundles on complex projective spaces, Progress in Mathematics, vol. 3, Birkhäuser, Boston, Mass., 1980. MR 561910
C. Peskine and L. Szpiro, Liaison des variétés algébriques. I, Invent. Math. 26 (1974), 271–302 (French). MR 364271, DOI 10.1007/BF01425554
C. Peterson, Applications of liaison theory to schemes supported on lines, growth of the deficiency module, and low rank vector bundles, Ph.D. thesis, Duke University (1994).
Sorin Popescu and Kristian Ranestad, Surfaces of degree $10$ in the projective fourspace via linear systems and linkage, J. Algebraic Geom. 5 (1996), no. 1, 13–76. MR 1358034
Dario Portelli and Walter Spangher, On the equations which are needed to define a closed subscheme of the projective space, J. Pure Appl. Algebra 98 (1995), no. 1, 83–93. MR 1317000, DOI 10.1016/0022-4049(95)90025-X
Kristian Ranestad, Surfaces of degree $10$ in the projective fourspace, Problems in the theory of surfaces and their classification (Cortona, 1988) Sympos. Math., XXXII, Academic Press, London, 1991, pp. 271–307. MR 1273383
A. Prabhakar Rao, Liaison among curves in $\textbf {P}^{3}$, Invent. Math. 50 (1978/79), no. 3, 205–217. MR 520926, DOI 10.1007/BF01410078
Prabhakar Rao, Liaison equivalence classes, Math. Ann. 258 (1981/82), no. 2, 169–173. MR 641822, DOI 10.1007/BF01450532
Riccardo Re, On the hyperplane sections of a projective algebraic variety, Matematiche (Catania) 42 (1987), no. 1-2, 211–218 (1989) (Italian, with English summary). MR 1030918
Peter Schenzel, Notes on liaison and duality, J. Math. Kyoto Univ. 22 (1982/83), no. 3, 485–498. MR 674606, DOI 10.1215/kjm/1250521732
Michael Schneider, Boundedness of low-codimensional submanifolds of projective space, Internat. J. Math. 3 (1992), no. 3, 397–399. MR 1163732, DOI 10.1142/S0129167X92000163
Charles A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994. MR 1269324, DOI 10.1017/CBO9781139644136