Boundedness for surfaces in weighted P^4

TL;DR

This paper establishes the existence of degree bounds for quasismooth non-general type surfaces in weighted projective 4-space, extending classical results to weighted settings and providing explicit bounds in specific cases.

Contribution

It introduces bounds on the degree of quasismooth non-general type surfaces in weighted P^4, generalizing previous unweighted bounds and calculating explicit bounds for simple weight configurations.

Findings

Bound on degree exists in weighted P^4 for quasismooth non-general type surfaces.

Explicit degree bounds are computed for specific weight cases.

Extends classical unweighted bounds to weighted projective spaces.

Abstract

Ellingsrud and Peskine (1989) proved that there exists a bound on the degree of smooth non general type surfaces in P^4. 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.