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.