Overview
- Offers a unique perspective on the theory of algebraic curves and surfaces, and of linear systems of hypersurfaces
- The classification of algebraic surfaces is given with full treatment of the P_{12}-theorem by Castelnuovo and Enriques
- Classical and modern methods from computational algebraic geometry are shown with applications in geometric modeling
Part of the book series: SISSA Springer Series (SISSASS, volume 4)
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About this book
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.
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Keywords
- Effective Methods in Algebraic Geometry
- Castelnuovo-Mumford Regularity
- Ruled and Rational Surfaces
- Fibrations of Surfaces Over Curves
- Linear Systems of Hypersurfaces
- Horace Method
- Base Locus Lemmas
- Mori Chamber Decompositions
- Elimination Matrices
- Intersection of Curves and sSurfaces
- Algebraic Curves and Surfaces
- Intersection Theory of Curves on a Surface
- Canonical Divisor, Plurigenera
- P_{12}- Classification Theorem
- Elliptic Fibrations
- Special Effect Varieties
- Positivity of Divisors
- Rational Curves and Surfaces
- Implicitization
- Fibers of Rational Maps
Table of contents (3 chapters)
Reviews
“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)
Authors and Affiliations
About the authors
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.
Bibliographic Information
Book Title: Algebraic Curves and Surfaces
Book Subtitle: A History of Shapes
Authors: Laurent Busé, Fabrizio Catanese, Elisa Postinghel
Series Title: SISSA Springer Series
DOI: https://doi.org/10.1007/978-3-031-24151-2
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023
Hardcover ISBN: 978-3-031-24150-5Published: 05 May 2023
Softcover ISBN: 978-3-031-24153-6Published: 06 May 2024
eBook ISBN: 978-3-031-24151-2Published: 03 May 2023
Series ISSN: 2524-857X
Series E-ISSN: 2524-8588
Edition Number: 1
Number of Pages: XIV, 205
Number of Illustrations: 1 b/w illustrations, 13 illustrations in colour
Topics: Algebra, Algebraic Geometry