Algebraic curves, the Brill and Noether way (Q2286623)

The book contains an introduction to the theory of complex algebraic curves, which is well suited for a postgraduate course in algebraic geometry, or even for a graduate course of high level. The book is reasonably self-contained, for students with some background on the theory of polynomial rings. More advanced basis of commutative algebra are briefly collected in section 3.6. The point of view of the author is based on the classical work of \textit{A. Brill} and \textit{M. Nöther} [Math. Ann. 7, 269--316 (1874; JFM 06.0251.01)]. The author mainly studies the theory of plane curves with ordinary singularities. Since any projective curve is birationally equivalent to a plane curve with ordinary singularities, the exposition covers the basis of the theory of all projective curves. After the first, introductory chapter on hypersurfaces, the author devotes the second chapter to the local theory of germs of plane curves. Branches are defined and studied in terms of the associated Puiseux series. The chapter contains the formal definition of intersection multiplicity of two branches. The third chapter is devoted to the study of intersections of plane curves. Combining the local theory with properties of the resultant of two polynomials, the author proves the Theorem of Bézout, and the AF+BG Theorem of M. Noether. The final chapter starts with an introduction to birational transformations and the proof that Cremona transormations of \(\mathbb P^2\) can turn any curve into a curve with ordinary singularities. Then, the author introduces divisors and general linear series on plane curves, starting with linear series cut by all curves of fixed degree. The main tool used through the chapter is the definition of adjoint linear series to curves with ordinary singularities. The author defines the genus of a curve and describes its basic properties, including proofs of the Riemann-Roch Theorem, the Clifford Theorem, and the Hurwitz formula. The analysis contained in the book, in addition to a good initial insight into algebraic geometry, can stimulate the reader to reconsider the classical Brill-Noether approach to the theory of curves, still suitable of application to problems on special linear series which remain open even now.