36 Pages

# 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.