Abstract
Hartshorne conjectured that there exists a finite number of families of
nongeneral type smooth surfaces in projective fourspace.
In 1989, this conjecture was positively solved by Ellingsrud and Peskine.
The exact bound for the degree is, however, still open.
The question concerning the exact bound motivates us to attack the
classification of nongeneraltype smooth surfaces of small degree.
The most important step for solving this classification problem is
to construct an example in each family of smooth nongeneraltype
surfaces in projective fourspace.
The main purpose of this talk is to construct a smooth rational surface
of degree 12. The construction of this surface is reduced to finding
a point of a 4codimensional subvariety M of the grassmaniann
G(10,4) of 4quotient spaces in a 10dimensional vector space.
Over a finite field with q elements, the probability for a point in G(10,4)
to lie in M is, therefore, about 1/q^4.
So we can expect to find such a point within a reasonable amount of time by
picking points at random over a small field.
Last Updated: March 14 2005 by Hirotachi Abo
