Algebraic geometry has always been an important and central research area of Mathematical Sciences, whose richness and variety of problems are well known. At the same time these well established features regularly create deep and strong interactions with other disciplines. This is true in the case of historically consolidated relations, like with theoretical physics, but also in several more recent situations. To this regard it will be enough to mention the growing influence, both theoretical and practical, of methods originating from Algebraic Geometry in the general field of the advanced technologies, where they are connected to sectors like complexity theory, information technology, robotics, biosciences and so on. Furthermore Algebraic Geometry undergoes today a remarkable expansion, solicited by the growth of new relevant directions of research, well represented in this network and in its programs, and boosted by the impressive progress of the last decade in the classification of algebraic varieties. These aspects display their very positive influence on the entire discipline. Finally the italian contribution to contemporary research in algebraic geometry is of primary importance at an international level. It is noteworthy that a relevant part of italian algebraic geometers will participate to the Project, so bringing in it strong competences and a variety of international scientific ties of the highest quality. Accordingly, we stress the effectiveness of such a project in involving young, brilliant scholars in a research activity of the same level. This research project refers to such cultural and scientific background, aiming at organizing and optimizing at best its strong potential.
- A)BIRATIONAL AND PROJECTIVE GEOMETRY
- 1. Projective algebraic geometry
- 2. Birational properties
- 3. Classification of algebraic varieties
- B) ALGEBRAIC CURVES
- 1. Syzygies of projective curves
- 2. Curves and their moduli
- C) ALGEBRAIC SURFACES
- 1. Families of curves on algebraic surfaces
- 2. Surfaces of general type
- D) HYPERKAHLER AND CALABI YAU GEOMETRY
- 1. Geometry of hyperkahler varieties
- 2. Moduli of sheaves on Calabi Yau varieties
- 3. Connections to Physics
- E) COMBINATORIAL ALGEBRAIC GEOMETRY
- 1. Toroidal compactifications
- 2. Tropicalization of moduli spaces
- F) EFFECTIVE METHODS WITH APPLICATIONS
- 1. Projective geometry of tensor decomposition
- 2. Identifiability
- 3. Some applications and interactions