Algebraic and Arithmetic Geometry Seminar

Fibrations in (1,2)-Surfaces

by Roberto Pignatelli (Trento)

Aula Seminari (Department of Mathematics)

Aula Seminari

Department of Mathematics


The content of this seminar stems from an ongoing collaboration with S. Coughlan, Y. Hu, and T. Zhang. By "(1,2)-surfaces" we denote complex algebraic surfaces with canonical singularities, ample canonical system, volume 1, and geometric genus 2. This is a class of surfaces that played a significant role in the theory of surfaces of general type in the last century and has shown in this century to also play an important role in the theory of 3-dimensional varieties, particularly in the recent proof of the 3-dimensional Noether inequality obtained by J. Chen, M. Chen, and C. Jiang. In fact, 3-dimensional varieties fibred in surfaces of type (1,2) play a role in this proof similar to that played by fibrations in curves of genus 2 in lower dimensions. In this seminar, I will introduce the concept of "simple" fibration in surfaces of type (1,2) and explain how through this concept we obtained a complete classification of 3-folds that satisfy the  equality in the aforementioned inequality. I will also discuss analogies and differences with the 2-dimensional case. Finally, I will mention some open problems that we are currently investigating.