Computations and bounds for surfaces in weighted projective four-spaces

Abstract

Many researchers have constructed examples of non general type surfaces in weighted projective spaces in various dimensions. Most of these constructions so far have been concentrated on complete intersections, and in the past three decades there has been a lot of success in this direction. Nowadays we have seen use of computer algebra systems to handle examples that are too cumbersome to do by hand. All smooth projective surfaces can be embedded in P5, but only few of them in P4. The most amazing fact is that the numerical invariants of any smooth surface in P4 must satisfy the double point formula. A natural question is whether there are any non general type surfaces in four dimensional weighted projective space, P4(w), which are not complete intersections. We believe that the answer is \yes, but they are not abundant". This thesis shows the �rst part, and justi�es the second part. That is, this thesis has two distinctive parts. First we prove that families of non general type surfaces in weighted projective four{space, P4(w) are rare by showing that their corresponding covers in straight P4, which are usually general type surfaces, are rare. In the second part we construct explicit examples of these rare objects in P4 using a technique involving sheaf cohomology and the Beilinson monad. We concentrate on the case of weights w = (1; 1; 1; 1; 2) for our particular examples. We present three explicit examples, one of which is symmetric. The main computer algebra system used is Macaulay2, Version 1.1 developed by D. Grayson and M. Stillman.

Date of Award	1 Oct 2009
Original language	English
Awarding Institution	University of Bath
Supervisor	Gregory Sankaran (Supervisor)