Boundedness for surfaces in weighted P

Ellingsrud and Peskine (1989) proved that there exists a bound on the degree of smooth non general type surfaces in P. The latest proven bound is 52 by Decker and Schreyer in 2000. In this paper we consider bounds on the degree of a quasismooth non-general type surface in weighted projective 4-space. We show that such a bound in terms of the weights exists, and compute an explicit bound in simple cases. Introduction Ellingsrud and Peskine [EP] proved that there exists an integer d0 such that all smooth non-general type surfaces in P4 have degree less than or equal to d0. This motivated a search for such surfaces, partly by computational methods, and also an effort to find an effective bound on d0, begun by Braun and Fløystad in [BF]. As far as we know the smallest proven bound is 52 by Decker and Schreyer [DS]. Some of the methods used to find such surfaces are also applicable to surfaces in weighted projective spaces P4(w) (some first steps in this direction are taken in [Ra]). It is therefore natural to ask whether a similar bound can be found for the degree of quasismooth non-general type surfaces in a weighted projective space with given weights. In this paper we show that such a readily computable bound (of course depending on the weights) does exist, and we compute it in some cases. To show that a bound exists all we need is a fairly simple adaptation of the way in which the results of [EP] (or [BF]) are applied. For a computable bound we use the results of [BF] together with some information about the contribution from the singularities of the surface in P4(w). Our procedure is to exploit the representation of P4(w) as a quotient of P4 by a finite group action. Starting with a quasismooth non-general type surface X in weighted projective 4-space P4(w), we take its cover in P4. This will (usually) be of general type, but it will have invariants bounded in terms of those of X, and the results of [BF] still apply in this situation.