Summary
This paper is dedicated to the study of Hilbert functions and Betti numbers of the projective varieties in a flat family. We prove that the Hilbert function H(X y ,n),y ∈ Y-a parameter scheme-is lower semicontinuous for any fixed n. In case Y is integral and noetherian we obtain the well-known fact that the set V ⊂Y where H(X y ,n)is maximal for all n's is open and nonempty. We show also that bi(X y )-the i- th Betti number of Xy—is upper semicontinuous for y ∈ V. The paper contains also a number of results concerning the relations among the various Betti numbers.
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A. V.Geramita - D.Gregory - L.Roberts,Monomial ideals and points in Projective Space, Queen's University Mathematical Preprint No. 1983-23.
A.Geramita - P.Maroscia,The ideal of forms vanishing at a finite set of points in Rn, J. of Alg. (to appear).
A. Geramita -F. Orecchia:Minimally generating ideals defining tangent cones, J. of Alg.,78 (1982), pp. 36–57.
R.Hartshorne,Algebraic Geometry, GTM 52, Springer-Verlag, 1977.
R. Hartshorne,Connectedness of the Hilbert scheme, Publ. Math. I.H.E.S. No.,29 (1966), pp. 261–304.
H.Matsumura,Commutative Algebra, Benjamin/Cummings Publ. Co., 1980.
D.Mumford - J.Fogarty,Geometric Invariant theory, Erg. der Math. No.,34, Springer-Verlag, 1982.
L. G. Roberts,A conjecture on Cohen-Macaulay type, C. R. Math. Rep. Acad. Sci. Canada,3 (1981), pp. 63–68.
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Member of G.N.S.A.G.A.-C.N.R. Supported in part by M.P.I. (Italian Minstry of Education).
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Boratyński, M., Greco, S. Hilbert functions and Betti numbers in a flat family. Annali di Matematica pura ed applicata 142, 277–292 (1985). https://doi.org/10.1007/BF01766597
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DOI: https://doi.org/10.1007/BF01766597