Algebraic Curves and Surfaces: A History of Shapes

ISBN-13: 9783031241505 / Angielski

Laurent Busé; Fabrizio Catanese; Elisa Postinghel

Opis

This volume collects the lecture notes of the school TiME2019 (Treasures in Mathematical Encounters). The aim of this book is manifold, it intends to overview the wide topic of algebraic curves and surfaces (also with a view to higher dimensional varieties) from different aspects: the historical development that led to the theory of algebraic surfaces and the classification theorem of algebraic surfaces by Castelnuovo and Enriques; the use of such a classical geometric approach, as the one introduced by Castelnuovo, to study linear systems of hypersurfaces; and the algebraic methods used to find implicit equations of parametrized algebraic curves and surfaces, ranging from classical elimination theory to more modern tools involving syzygy theory and Castelnuovo-Mumford regularity. Since our subject has a long and venerable history, this book cannot cover all the details of this broad topic, theory and applications, but it is meant to serve as a guide for both young mathematicians to approach the subject from a classical and yet computational perspective, and for experienced researchers as a valuable source for recent applications.

Recenzje

"The geometry of algebraic curves and surfaces is a wide and venerable subject, and there are many monographs and textbooks aimed to graduate students and experienced researchers. ... The book can be used as a gentle introduction to the theory of algebraic surfaces in conjunction with more systematic textbooks such as C. Ciliberto's Classification of Complex Algebraic Surfaces." (Felipe Zaldivar, MAA Reviews, July 30, 2023)

Spis treści

Chapter 1. The $P_$-Theorem: the Classification of surfaces and its historical development, written by Fabrizio Catanese.

The first chapter explains the main steps and the strategy of the classification of algebraic surfaces, with a view to higher dimensional geometry, and with a historical discussion of the achievement of the classification, starting from the work of Castelnuovo and Enriques and encompassing the many developments which took place in the 20th century. It contains the first full treatment of the so-called P_-theorem by Castelnuovo and Enriques, including a novel critical analysis.

Chapter 2. Linear systems of hypersurfaces and beyond, written by Elisa Postinghel.

The second chapter treats the dimensionality problem for linear systems of plane curves and of hypersurfaces of projective n-spaces with assigned multiple points. It offers a survey on the progress made towards this and related questions over a 120 year time span. The ideas presented stem from work of G. Castelnuovo and of B. Segre and are coupled with more modern tools from toric geometry, the theory of Mori dream spaces and sheaf cohomology.

Chapter 3. Implicit representations of algebraic curves and surfaces, written by Laurent Busé.

The last chapter deals with a basic issue in geometry: one may easily find the intersection of a curve with a surface in three dimensional space if the surface is given via an implicit description through an equation. If the surface is given parametrically, the solution of the problem goes through the implicitization of the surface. This problem belongs to the classical elimination theory, and has seen an impetuous development in the last 30 years through new methods from homological and commutative algebra.

O autorze

Laurent Busé received his PhD degree in Mathematics at the Université of Nice - Sophia Antipolis in 2001 and he is currently a senior researcher at the Inria research center of Université Côte d’Azur. His main research interests focus on computational methods in algebraic geometry and commutative algebra, more specifically on elimination theory, the geometry of algebraic curves and surfaces and their applications in the fields of geometric modeling and geometry processing.

Fabrizio Catanese studied at the Universita’ di Pisa and Scuola Normale Superiore 1968-1974, held the Chair of Geometry 1980-1997 in Pisa, the Gauss Chair of Complex Analysis in Goettingen, 1997-2001, then has been professor in Bayreuth since 2001. He is Research Scholar at the Korean Institute for Advanced Study and member of the Accademia Nazionale dei Lincei, the Goettingen Academy, the Academia Europaea. He has been visiting professor at many international Universities and research centres. 

Elisa Postinghel received her PhD in Mathematics from the University Roma Tre in 2010. She was a lecturer at Loughborough University between 2016 and 2020 and she is currently a senior researcher at the University of Trento. Her research work is in algebraic geometry; her main interests span classical topics such as polynomial interpolation problems on higher dimensional varieties as well as birational geometry and positivity properties of divisors and curves on Mori dream spaces.