Hartshorne conjectured that there exists a finite number of families of smooth rational 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 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 4-codimensional subvariety M of the grassmaniann G(10,4) of 4-quotient spaces in a 10-dimensional 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: October 21 2004 by Hirotachi Abo