Inequidimensionality of Hilbert schemes

Abstract

We give a lower bound on the number of distinct dimensions of irreducible components of the Hilbert scheme of codimension 2 subvarieties in P1W, for n < 5 (respectively, the moduli space of surfaces or 3-folds) in terms of the Hilbert polynomial (resp. Chern numbers). Let Hilbp be the Hilbert scheme of subvarieties in the projective space with fixed Hilbert polynomial P (respectively, let M be a moduli space of varieties with fixed Chern numbers). It is known that Hilbp (resp. M) has finitely many irreducible components and that the number of these components is bounded by some function of the Hilbert polynomial (resp. the Chern numbers). For work on the number of components of the Hilbert scheme (resp. the moduli space), see [EHM] for curves in P3 and [Chl] for codimension 2 subvarieties in Ipn with n ?5 (resp. [Cal], [Ca2], [Ca3], [M] for surfaces and [Chl] for surfaces and 3-folds). The next question to ask is whether the Hilbert scheme (resp. moduli space) is equidimensional if it is reducible. Catanese [Ca3] has shown that for M, the moduli space of surfaces, the number of distinct dimensions can be arbitrarily large. In this note we study the number of distinct dimensions of the components of the Hilbert scheme Hilbp (resp. moduli space M) parametrizing subschemes with intersection numbers H'Kn-2-i (resp. Chern numbers), where H is the hyperplane class and K is the canonical class. We define n(d, HK, K2) = #{dimHIH is a component of the Hilbert scheme of surfaces in JR'4 with intersection numbers d, HK, K2 }, n(d, H2K, HK2, K3) = #{dim HIH is a component of the Hilbert schemes of 3-folds in I5 with intersection numbers d, H2K, HK2, K3}, n(K2, C2) = #{dim HIH is a component of the moduli space of surfaces with Chern numbers K2, c21, n(K3,c1c2,c3) = #{dimHIH is a component of the moduli space of 3-folds with Chern numbers K3, C1 C2, C3 }. Received by the editors October 5, 1995 and, in revised form, March 14, 1996. 1991 Mathematics Subject Classification. Primary 14J29; Secondary 14M07, 14M12.