ncts

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.