Even Linkage Classes

Nollet, Scott

doi:10.1090/S0002-9947-96-01552-8

Even Linkage Classes
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by Scott Nollet

Trans. Amer. Math. Soc. 348 (1996), 1137-1162

DOI: https://doi.org/10.1090/S0002-9947-96-01552-8

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Abstract:

In this paper we generalize the $\mathcal {E}$ and $\mathcal {N}$-type resolutions used by Martin-Deschamps and Perrin for curves in $\mathbb {P}^{3}$ to subschemes of pure codimension in projective space, and shows that these resolutions are interchanged by the mapping cone procedure under a simple linkage. Via these resolutions, Rao’s correspondence is extended to give a bijection between even linkage classes of subschemes of pure codimension two and stable equivalence classes of reflexive sheaves $\mathcal {E}$ satisfying $H^{1}_{*}( \mathcal {E})=0$ and $\mathcal {E}xt^{1}( \mathcal {E}^{\vee }, \mathcal {O})=0$. Further, these resolutions are used to extend the work of Martin-Deschamps and Perrin for Cohen-Macaulay curves in $\mathbb {P}^{3}$ to subschemes of pure codimension two in $\mathbb {P}^{n}$. In particular, even linkage classes of such subschemes satisfy the Lazarsfeld-Rao property and any minimal subscheme for an even linkage class links directly to a minimal subscheme for the dual class.

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
Giorgio Bolondi and Juan C. Migliore, The structure of an even liaison class, Trans. Amer. Math. Soc. 316 (1989), no. 1, 1–37. MR 968882, DOI 10.1090/S0002-9947-1989-0968882-2
A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III, Inst. Hautes Études Sci. Publ. Math. 28 (1966), 255. MR 217086
Robin Hartshorne, Ample subvarieties of algebraic varieties, Lecture Notes in Mathematics, Vol. 156, Springer-Verlag, Berlin-New York, 1970. Notes written in collaboration with C. Musili. MR 0282977, DOI 10.1007/BFb0067839
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157, DOI 10.1007/978-1-4757-3849-0
Robin Hartshorne, Stable reflexive sheaves, Math. Ann. 254 (1980), no. 2, 121–176. MR 597077, DOI 10.1007/BF01467074
R. Hartshorne, Generalized Divisors on Gorenstein Schemes, K-Theory 8 (1994), 287-339.
Steven L. Kleiman, Geometry on Grassmannians and applications to splitting bundles and smoothing cycles, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 281–297. MR 265371, DOI 10.1007/BF02684605
Robert Lazarsfeld and Prabhakar Rao, Linkage of general curves of large degree, Algebraic geometry—open problems (Ravello, 1982) Lecture Notes in Math., vol. 997, Springer, Berlin, 1983, pp. 267–289. MR 714753, DOI 10.1007/BFb0061648
Mireille Martin-Deschamps and Daniel Perrin, Sur la classification des courbes gauches, Astérisque 184-185 (1990), 208 (French). MR 1073438
M. Martin-Deschamps and D. Perrin, Construction de Courbes Lisses: un Théorème de Bertini, vol. 22, Laboratoire de Mathématiques de l’Ecole Normale Supérieure, 1992.
Hideyuki Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid. MR 879273
J. Migliore, An Introduction to Deficiency Modules and Liaison Theory for Subschemes of Projective Space, Global Analysis Research Center, Seoul National University, Lecture Notes Series 24 (1994).
C. Peskine and L. Szpiro, Liaison des variétés algébriques. I, Invent. Math. 26 (1974), 271–302 (French). MR 364271, DOI 10.1007/BF01425554
Dunham Jackson, A class of orthogonal functions on plane curves, Ann. of Math. (2) 40 (1939), 521–532. MR 80, DOI 10.2307/1968936
Prabhakar Rao, Liaison equivalence classes, Math. Ann. 258 (1981/82), no. 2, 169–173. MR 641822, DOI 10.1007/BF01450532