Noether-Lefschetz theory and class groups (John Brevik, Cal State Long Beach)

The classical Noether-Lefschetz Theorem states that a suitably general algebraic surface S of degree d ≥ 4 in complex projective 3-space P3 contains no curves besides complete intersections, that is, curves of the form S ∩ T where T is another surface. After discussing briefly Noether’s non-proof of this theorem and hinting at the idea behind Lefschetz’s proof, I will sketch some of our recent progress in generalizing this theorem and its implications for global and local divisor class groups. We explore the question of what class groups are possible for local rings on surfaces in a particular analytic isomorphism class and show the ubiquitousness of unique factorization domains among such rings. Joint work with Scott Nollet (Texas Christian University).