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Colloquium, Spring 2006

Day: Monday, March 20th Time: 3:00pm Place: BH201 Title: Finding Needles in a Haystack: Theory and Computation in the Search for Special Surfaces Speaker:Hirotachi Abo (Colorado State) Abstract: Algebraic geometry is, roughly speaking, the study of solutions to systems of polynomial equations. Such systems arise very naturally throughout Mathematics, Computer Science and Engineering. The fundamental geometric objects determined by systems of polynomials are called (algebraic) varieties. One of the most important problems in algebraic geometry is the classification of varieties. Some basic classification questions are: (1) Does there exist a variety with a given set of invariants? (2) How big is the parameter space of such varieties? (3) How can such varieties be constructed explicitly? To solve such problems requires a mixture of theory, computation and experimentation. In this talk, we will start by discussing the relationship between algebraic varieties and ideals. Then we will go through some basics of Groebner basis theory and the role of computation in algebraic geometry. We will illustrate these ideas with a variety arising in mechanical engineering and with an elementary example from linear algebra. At the end of the talk, I will discuss a proof of the existence of smooth rational surfaces of degree 12 in projective fourspace. These surfaces were found by a random search over a finite field employing a ``needle in a haystack" approach. We use theory to guide us to the right ``haystack", we use computation to search the haystack (over a finite field) and theory enters again in allowing us to lift the result to the complex numbers. The surfaces found by this method are new and of interest in the classification of smooth non-general type surfaces in projective fourspace. This classification problem is motivated by the theorem of Ellingsrud and Peskine, which says that the degree of smooth non-general type surfaces in projective fourspace is bounded. Rational surfaces in projective fourspace are of non-general type and were previously only known to exist in degrees up to 11. Our result establishes the existence of families of degree 12, is constructive, and sets a new degree record.