Algebraic Curves and Surfaces

Beschreibung

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.

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.
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

Autor*in

Laurent Busé

Themen in »Algebraic Curves and Surfaces«

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

Stimmen zu »Algebraic Curves and Surfaces«

“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)

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