Jan O. Kleppe
Mathematics Subject Classification (1991): 14C05, 14J10, 14B10, 14B15, 13D45
INTRODUCTION.
In the present paper we study the Hilbert scheme H(d,p,\pi) of all surfaces of degree d and arithmetic (resp. sectional) genus p (resp. \pi). Recall that, for space curves C in P3 = Proj(R), Martin-Deschamps and Perrin have given a stratification H(d,g)\gamma,\rho of the Hilbert scheme H(d,g) of space curves of degree d, genus g obtained by deforming curves with constant cohomology [MDP1]. They also proved the smoothness of the "morphism" f : H(d,g)\gamma,\rho -> E\rho = isomorphism classes of R-modules M of finite length, given by (C \subset P3) -> M : = H*1(IC(v)), they gave a scheme structure to H(d,g)\gamma,M : = f-1(M) and computed its dimension. Earlier Rao proved that any R-module M of finite length determines the liaison class of a curve C, up to a shift in the grading. Note that Rao's result is related to the surjectivity of f, while the smoothness implies infinitesimal surjectivity as well. For surfaces in P4 there is a recent result of Bolondi [B2], similar to that of Rao, telling that a triple D = (M1,M2,b) of modules Mi of finite length and an extension b of 0Ext2(M2, M1) determines the liason class of a surface X such that Mi \simeq H*i(IX(v)) modulo some shift in the grading. Therefore it is natural to consider the stratification H\gamma,\rho : = H(d,p,\pi)\gamma,\rho of H(d,p,\pi), similar to the one in the curve case, and to ask if the corresponding f : H\gamma,\rho -> V\rho = isomorphism classes of R-modules M1 and M2 commuting with b, is smooth and irreducible. We prove in this paper that the answer is yes (theorem 1.1), thus extending Bolondi's result in this direction. It follows that the fiber H(d,p,\pi)\gamma,D : = f-1(D) is smooth and irreducible and we compute its dimension (corollary 2.7). In section 3 we also determine the tangent space of H\gamma,\rho (resp. V\rho) at (X \subset P4) (resp. at D), from which we deduce a local isomorphism H\gamma,\rho \simeq H(d,p,\pi) (theorem 3.7) under certain restrictive conditions (e.g. natural cohomology) and a smoothness criterion for V\rho (proposition 3.4). The liaison result we prove in theorem 4.1 turns out to be helpful in determining the structure of H\gamma,\rho and its dimension. Note that the irreducibility of H\gamma,D follows from an earlier work of Bolondi [B1] while, in the special case of arithmetically Cohen-Macaulay surfaces (i.e. surfaces with Mi = 0 for i = 1,2), both the irreducibility and the smoothness of H\gamma,D follow from [E]. To see more generally how H\gamma,\rho determine H(d,p,\pi), it is desireable to study the imbedding H\gamma,\rho-> H(d,p,\pi) in detail as we did in [K3] for the Hilbert scheme of curves. The corresponding problem for H(d,p,\pi) will eventually be carried out in another paper. Indeed as we will see in what follows, the technical problems in describing the stratification of H(d,p,\pi), the tangent spaces of H\gamma,\rho and V\rho etc. are much more complicated than in the curve case, justifying this limitation.
We also limit the extent of this paper by omitting proving that H(d,p,\pi)\gamma,D is a scheme, although it seems quite natural to generalize the work of [MDP1] so far. Indeed the "morphism" f : H\gamma,\rho -> V\rho to the "scheme" (i.e. stack) V\rho has a natural nice description in terms of the hulls of the local deformation functors at a given point (X \subset P4). In this local case H(d,p,\pi)\gamma,D corresponds to the hull of the local fiber functor. Even though we have not proved the existence of all this as schemes, we allow a thinking and a terminology as if they were schemes, knowing that the statements have a precise interpretation in terms of their corresponding "completed local rings", i.e. their hulls. Only the irreducibility is problematic from this local point of view, but this case is already taken care of in the literature by [B1] and [BM1]. To limit the size of this paper, we only sketch the proof of some other results (e.g the tangent spaces of H\gamma,\rho and V\rho) as well.
Due to the importance of the works of Martin-Deschamps and Perrin and its consequences for the Hilbert scheme H(d,g) of curves ([MDP1], [MDP2]), we hope the corresponding theory for the Hilbert scheme H(d,p,\pi) of surfaces, of which we take a first main step, will turn fruitful.
The investigations of this paper started several years ago as a common project with prof. G. Bolondi at Sassari. It was prof. G. Bolondi who introduced me to the idea of extending the results of [B2], and who pointed out the interesting things to be proved. As time has pasted it is the author of this paper who has carried out the investigations, and as agreed upon by Bolondi, should be the sole author of this paper. I thank prof. G. Bolondi very much for many stimulating discussions while preparing this work and prof. E. Ballico and the University of Trento for their hospitality during my visits in June 1994 and May 1995. This paper was written in the context of EUROPROJ.