The overall objective of the project is the research in one of the crucial areas of mathematics, namely algebraic geometry. Its purpose is to contribute to a significant growth of knowledge on algebraic varieties, studying their general properties, the specific families they form, the rich and thriving geometry they reveal. The research will investigate a wide range of central questions, distributed almost all over the field, with the aim of reaching effective progress and new relevant answers. A variety of distinct competences and techniques, as typical in algebraic geometry, will be used to address the problems in their widest perspective. With a complementary attitude, the attention will also concentrate on a number of concrete examples of outstanding geometric relevance. These, in a way, are the true leading threads of the research and crucial for understanding its trends. The joint work of the units will focus on them. Furthermore, a remarkable trend of today’s research in algebraic geometry is represented by the so called effective methods. One can say that these represent one of the bridges connecting pure research in algebraic geometry to a wide spectrum of technological applications, for instance Signal Processing, Computer Vision, Complexity Theory, Web Search and so on. Effective methods naturally fit in this project, a section of it is dedicated to them and to their not disjoint, concrete or potential, applications: see part B 4. The project 'Geometry of Algebraic Varieties' reflects the scientific features and the character of the Italian contemporary algebraic geometry. It includes a relevant and strong component of this School, with many of the principal scholars. A number of previous PRIN projects were successfully run in the same network, performing top research results along the years. The research problems to be considered by this project belong to the following thematic sections A, B, C, D, E, F: 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 decompostion. 2. Identifiability. 3. Some applications and interactions.
PRIN 2015 - Geometry of Algebraic Varieties
Mella, Massimiliano;
2017
Abstract
The overall objective of the project is the research in one of the crucial areas of mathematics, namely algebraic geometry. Its purpose is to contribute to a significant growth of knowledge on algebraic varieties, studying their general properties, the specific families they form, the rich and thriving geometry they reveal. The research will investigate a wide range of central questions, distributed almost all over the field, with the aim of reaching effective progress and new relevant answers. A variety of distinct competences and techniques, as typical in algebraic geometry, will be used to address the problems in their widest perspective. With a complementary attitude, the attention will also concentrate on a number of concrete examples of outstanding geometric relevance. These, in a way, are the true leading threads of the research and crucial for understanding its trends. The joint work of the units will focus on them. Furthermore, a remarkable trend of today’s research in algebraic geometry is represented by the so called effective methods. One can say that these represent one of the bridges connecting pure research in algebraic geometry to a wide spectrum of technological applications, for instance Signal Processing, Computer Vision, Complexity Theory, Web Search and so on. Effective methods naturally fit in this project, a section of it is dedicated to them and to their not disjoint, concrete or potential, applications: see part B 4. The project 'Geometry of Algebraic Varieties' reflects the scientific features and the character of the Italian contemporary algebraic geometry. It includes a relevant and strong component of this School, with many of the principal scholars. A number of previous PRIN projects were successfully run in the same network, performing top research results along the years. The research problems to be considered by this project belong to the following thematic sections A, B, C, D, E, F: 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 decompostion. 2. Identifiability. 3. Some applications and interactions.I documenti in SFERA sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
