Blowup and specialization methods for the study of linear systems

The computation of the dimension of linear systems of curves with imposed base multiple points on surfaces is a difficult problem, with open conjectures that are being approached only with partial success. Among others, blowup-based techniques and degenerations show some promise of leading to satisfactory answers. We present an overview of such blowup based techniques at an introductory level, with emphasis on clusters of infinitely near points and Ciliberto-Miranda's blowup and twist.