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.

**Laurent Busé** is a senior researcher at the Inria research center of Université Côte d’Azur

**Fabrizio Catanese** is Research Scholar at the Korean Institute for Advanced Study

**Elisa Postinghel** is professor at the Department of Mathematics of the University of Trento

**From the Preface (pagg. V-IX)**

[…]

TiME2019 was inspired by the deep work of Castelnuovo and Enriques, two of the main figures of the Italian school of Algebraic Geometry. […] Algebraic geometry studies the shape and the properties of the zero set of a system of polynomial equations, called algebraic variety. […] Curves and surfaces are the low-dimensional algebraic varieties.

[…]

Complex smooth curves were already studied after the fundamental work of Riemann, and their classification according to their genus was already understood. Algebraic surfaces instead resisted many attempts to their classification. In Rome, Castelnuovo met the young Enriques […]. Their collaboration started a new approach to the study of algebraic surfaces via the analysis of the families of algebraic curves contained in them.

[…]

The classification entailed the introduction of a vast number of results and geometric techniques. The results and methods inspired and guided most of the work of the Italian school until the 1940s with the hope of studying also higher dimensional varieties. […] The ideas of Castelnuovo and Enriques and their geometric intuitions can now be found in several currently active research areas.

[…]

The first goal of TiME2019 has this in mind: to make young researchers aware that, in original classical papers, we can find mathematical gems with ideas and intuitions which are still of current interest and can still drive new original research. The second goal is to underline how the questions and interests about some mathematical object can vary through time, by giving different perspectives on the same objects.

For instance, many modern applications make use of algebraic curves and surfaces; for instance, in geometric modeling, 3D shapes are often represented by rational curves and surfaces. This motivates the search for efficient methods to reach optimal descriptions of these models as solutions of polynomial systems, and the underlying theory and the accumulated knowledge provide invaluable tools.

[…]

The volume is divided into the following chapters.

Chapter 1. The P12-Theorem: The Classification of Surfaces and Its Historical Development by Fabrizio Catanese, U. Bayreuth, Germany.

The classification theorem of algebraic surfaces via the 12-th plurigenus P12 was achieved by Castelnuovo and Enriques in 1914: it divides surfaces into birational equivalence classes […]. The purpose of these lectures is first to explain the statement and the strategy of the classification theorem, then to state and prove these main results in the language of the mathematics of the twentieth century. […] This characterization was not fully achieved by Enriques, and we discuss in the appendix the cases that they were missing.

Chapter 2. Linear Systems of Hypersurfaces and Beyond by Elisa Postinghel, U. di Trento, Italy.

This chapter deals with linear systems of plane curves and of hypersurfaces of complex projective spaces of arbitrary dimension, with assigned multiplicity at a given collection of points. The main question is to determine the dimension of these linear systems and, aside from special cases, it is still a widely open question. […] The main tools to tackle these questions […] originally appeared in work of the Italian school of algebraic geometry and, in particular, of Guido Castelnuovo and of Beniamino Segre. […]

Chapter 3. Implicit Representations of Algebraic Curves and Surfaces by Laurent Busé, U. Côte d’Azur, Inria, France.

The topic of this chapter is the implicitization of rational algebraic curves and surfaces in projective spaces, that is the computation of the equation(s) of the image of curve and surface parametrizations. This is […] an important subject in elimination theory, that has seen a renewed interest during the last thirty years, largely because of its usefulness in geometric modeling. […] The purpose of this chapter is to explore how methods from algebraic geometry and commutative algebra have been used and developed over the years to solve the implicitization problem […].

Courtesy by Springer