Springer Ergebnisse der Mathematik und ihrer Grenzgebiete 61

Algebraic Geometry

Chapter I. Theory and Reduction of Singularities Edit

Algebraic varieties and birational transformations Edit

  • Equation (2): Let's unpack this. It's claimed that, for an irreducible projective variety $ V \subset \mathbb{P}^r $, there exists a hypersurface $ H = V(\varphi) \subset \mathbb{P}^{k+1} $ such that, for an appropriate choice of coordinates on $ \mathbb{P}^r $, there is a map $ H \rightarrow \mathbb{P}^r $ sending $ [t_1 : \cdots : t_{k+2}] $ to $ \left[\frac{\partial \varphi}{\partial t_1} t_1 : \frac{\partial \varphi}{\partial t_1} t_2 : \cdots : \frac{\partial \varphi}{\partial t_1} t_{k+2} : \psi_1\left(\vec{t}\right) : \cdots  :\psi_{r-k-1} \left(\vec{t}\right)\right] $ for some homogeneous polynomials $ \psi_i $ which is a dominant map to V. That is, this is some sort of implicit function theorem.

Singularities of plane algebraic curves Edit

Singularities of space algebraic curves Edit

Topological classification of singularities Edit

Singularities of algebraic surfaces Edit

The reduction of singularities of an algebraic surface Edit

Chapter II. Linear Systems of Curves Edit

Chapter III. Adjoint Systems and the Theory of Invariants Edit

Chapter IV. The Arithmetic Genus and the Generalized Theorem of Riemann-Roch Edit

Chapter V. Continuous Non-linear systems Edit

Chapter VI. Topological properties of algebraic surfaces Edit

Chapter VII. Simple and double integrals on an Algebraic Surface Edit

Chapter VIII. Branch Curves of Multiple Planes and Continuous Systems of Plane Algebraic Curves Edit

Appendix A. A Series of Equivalence Edit

Appendix B. Correspondences between Algebraic Varieties Edit