Zariski - Algebraic Surfaces

Springer Ergebnisse der Mathematik und ihrer Grenzgebiete 61

Algebraic Geometry

= Chapter I. Theory and Reduction of Singularities =

Algebraic varieties and birational transformations

 * 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.

The reduction of singularities of an algebraic surface
= Chapter II. Linear Systems of Curves =

= Chapter III. Adjoint Systems and the Theory of Invariants =

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

= Chapter V. Continuous Non-linear systems =

= Chapter VI. Topological properties of algebraic surfaces =

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

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

= Appendix A. A Series of Equivalence =

= Appendix B. Correspondences between Algebraic Varieties =