Algebraic curves and surfaces. A history of shapes (Q6046399)

The book contains the notes of three courses given by the authors during the summer school for young researchers TiME2019, held in Levico Terme (Italy). In the first section, the second author outlines the classification of projective surfaces \(S\), described by Castelnuovo and Enriques at the beginning of the XXth century. The classification is based on the Kodaira dimension \(\kappa(S)\), which corresponds to the dimension of the image of the map associated with \(mK_s\) for \(m\gg 0\). The case \(\kappa(S)=2\) corresponds to surfaces of general type, whose moduli spaces are still intensively studied. The case of rational and ruled surfaces is studied in the first two lectures. Then, the exposition focuses on the \(P_{12}\)-Theorem of Castelnuovo-Enriques, which shows how the birational structure of \(S\) depends on the dimension of the linear system \(12K_S\). In particular, when \(P_{12}>1\) and \(K_S^2=0\), the surface has a fibration \(S\to B\) over a curve, and the fibers are smooth elliptic curves. The classification of this case is based on the isotriviality of the fibration, i.e. the property that the elliptic fibers are isomorphic or, equivalently, their moduli are trivial. In the last lecture, several strategies for the proof of isotriviality are collected and illustrated. In the second section of the book the third author illustrates the main techniques and some recent achievements in the study of polynomial interpolation with multiplicities. The problem is to determine the dimension of linear systems of polynomials vanishing at \(k\) general points with multiplicities greater or equal than preassigned values. Even in the case of homogeneous polynomials in three variables over the complex field (corresponding to curves in the complex projective plane) the situation is not totally understood. A general method for the construction of linear systems whose dimension is bigger than the expected value is known; it is based on the geometry of the blow up of the plane at several general points. Long ago it has been conjectured that the method exhausts all the cases in which the dimension is bigger than expected. The conjecture is still open. The book contains a discussion on the conjecture, with a description of the cases in which it is known to hold, and an illustration of the main methods used to attack the problem (as Cremona transformations, and the study of the nef cone of blow up's). Possible extensions of the conjecture to higher dimensional spaces are also presented. The third section contains an outline of algebraic and geometric methods for the computation of implicit (polynomial) equations that describe algebraic varieties defined by parametric polynomials (hence defined as the image of suitable polynomial maps). The construction of implicit equations can provide a way to solve some problems, like the membership problem, relevant for applications. The straightforward method to obtain implicit equations is based on the classical elimination theory. Yet, when the number of variables and parameters increases, a direct use of elimination theory becomes unpractical. The first author describes a series of algebraic tools that can simplify the problem. These tools are based on a matricial representation of the associated ideals, and the author introduces and explains the role of the corresponding elimination matrices. The main application discussed in the section concerns implicit equations that define the intersection of parametric (rational) hypersurfaces, a problem that arises naturally in the reconstruction of images.