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 |
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Original language | English |
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Awarding Institution | |
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Supervisor | Gregory Sankaran (Supervisor) |
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Computations and bounds for surfaces in weighted projective four-spaces

Rammea, L. (Author). 1 Oct 2009

Student thesis: Doctoral Thesis › PhD