On derived categories of algebraic surfaces constructed via Q-Gorenstein smoothings

Abstract

This thesis mainly consists of the study of line bundles on algebraic surfaces constructed via $\BbbQ$ -Gorenstein smoothing. Recently, $\BbbQ$ -Gorenstein smoothing is used to construct new algebraic surfaces, however, it is relatively unknown how the geometric information varies along the smoothing. Hacking's weighted blow up technique, which has been used to construct exceptional vector bundles on a surface from its singular degenerations, is applied to study the line bundles on the surfaces obtained by $\BbbQ$ -Gorenstein smoothing. The main idea is to lift information about divisors to a (possibly non-Cartier) divisors on the central fiber of the smoothing. It turns out that numerical invariants of divisors, such as holomorphic Euler characteristics, can be understood via information from the central fiber. Also, the cohomological properties of the divisors can be studied with the same method, even if it is less efficient in general. In some nice and simple examples of $\BbbQ$ -Gorenstein smoothings, cohomologies of divisors are efficiently controlled under this method. As an example, cohomological properties of divisors in Dolgachev surfaces of type (2,3) will be studied in details. This leads to the presentation of semiorthogonal decompositions of the derived category into exceptional collection of 12 line bundles and a phantom category. Using the same method, we study the nonminimal Enriques surfaces, and establish semiorthogonal decompositions of the derived categories into 13 line bundles and quasi-phantom categories.