Event Type:

Department Colloquium

Date/Time:

Thursday, February 24, 2005 - 08:00

Location:

Kidd 364 (<b>NOTE: unusual day and time</b>)

Guest Speaker:

Institution:

Colorado State University

Abstract:

Hartshorne conjectured that there exists a finite number of families of non-general 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 nongeneral-type smooth surfaces of small degree. The most important step for solving this classification problem is, therefore, to construct an example in each family of smooth nongeneral-type 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 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.